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section\<open>Homology, II: Homology Groups\<close>
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theory Homology_Groups
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imports Simplices "HOL-Algebra.Exact_Sequence"
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begin
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subsection\<open>Homology Groups\<close>
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text\<open>Now actually connect to group theory and set up homology groups. Note that we define homomogy
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groups for all \emph{integers} @{term p}, since this seems to avoid some special-case reasoning,
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though they are trivial for @{term"p < 0"}.\<close>
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definition chain_group :: "nat \<Rightarrow> 'a topology \<Rightarrow> 'a chain monoid"
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where "chain_group p X \<equiv> free_Abelian_group (singular_simplex_set p X)"
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lemma carrier_chain_group [simp]: "carrier(chain_group p X) = singular_chain_set p X"
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by (auto simp: chain_group_def singular_chain_def free_Abelian_group_def)
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lemma one_chain_group [simp]: "one(chain_group p X) = 0"
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by (auto simp: chain_group_def free_Abelian_group_def)
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lemma mult_chain_group [simp]: "monoid.mult(chain_group p X) = (+)"
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by (auto simp: chain_group_def free_Abelian_group_def)
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lemma m_inv_chain_group [simp]: "Poly_Mapping.keys a \<subseteq> singular_simplex_set p X \<Longrightarrow> inv\<^bsub>chain_group p X\<^esub> a = -a"
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unfolding chain_group_def by simp
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lemma group_chain_group [simp]: "Group.group (chain_group p X)"
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by (simp add: chain_group_def)
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lemma abelian_chain_group: "comm_group(chain_group p X)"
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by (simp add: free_Abelian_group_def group.group_comm_groupI [OF group_chain_group])
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lemma subgroup_singular_relcycle:
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"subgroup (singular_relcycle_set p X S) (chain_group p X)"
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proof
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show "x \<otimes>\<^bsub>chain_group p X\<^esub> y \<in> singular_relcycle_set p X S"
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if "x \<in> singular_relcycle_set p X S" and "y \<in> singular_relcycle_set p X S" for x y
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using that by (simp add: singular_relcycle_add)
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next
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show "inv\<^bsub>chain_group p X\<^esub> x \<in> singular_relcycle_set p X S"
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if "x \<in> singular_relcycle_set p X S" for x
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using that
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by clarsimp (metis m_inv_chain_group singular_chain_def singular_relcycle singular_relcycle_minus)
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qed (auto simp: singular_relcycle)
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definition relcycle_group :: "nat \<Rightarrow> 'a topology \<Rightarrow> 'a set \<Rightarrow> ('a chain) monoid"
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where "relcycle_group p X S \<equiv>
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subgroup_generated (chain_group p X) (Collect(singular_relcycle p X S))"
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lemma carrier_relcycle_group [simp]:
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"carrier (relcycle_group p X S) = singular_relcycle_set p X S"
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proof -
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have "carrier (chain_group p X) \<inter> singular_relcycle_set p X S = singular_relcycle_set p X S"
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using subgroup.subset subgroup_singular_relcycle by blast
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moreover have "generate (chain_group p X) (singular_relcycle_set p X S) \<subseteq> singular_relcycle_set p X S"
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by (simp add: group.generate_subgroup_incl group_chain_group subgroup_singular_relcycle)
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ultimately show ?thesis
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by (auto simp: relcycle_group_def subgroup_generated_def generate.incl)
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qed
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lemma one_relcycle_group [simp]: "one(relcycle_group p X S) = 0"
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by (simp add: relcycle_group_def)
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lemma mult_relcycle_group [simp]: "(\<otimes>\<^bsub>relcycle_group p X S\<^esub>) = (+)"
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by (simp add: relcycle_group_def)
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lemma abelian_relcycle_group [simp]:
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"comm_group(relcycle_group p X S)"
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unfolding relcycle_group_def
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by (intro group.abelian_subgroup_generated group_chain_group) (auto simp: abelian_chain_group singular_relcycle)
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lemma group_relcycle_group [simp]: "group(relcycle_group p X S)"
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by (simp add: comm_group.axioms(2))
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lemma relcycle_group_restrict [simp]:
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"relcycle_group p X (topspace X \<inter> S) = relcycle_group p X S"
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by (metis relcycle_group_def singular_relcycle_restrict)
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definition relative_homology_group :: "int \<Rightarrow> 'a topology \<Rightarrow> 'a set \<Rightarrow> ('a chain) set monoid"
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where
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"relative_homology_group p X S \<equiv>
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if p < 0 then singleton_group undefined else
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(relcycle_group (nat p) X S) Mod (singular_relboundary_set (nat p) X S)"
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abbreviation homology_group
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where "homology_group p X \<equiv> relative_homology_group p X {}"
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lemma relative_homology_group_restrict [simp]:
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"relative_homology_group p X (topspace X \<inter> S) = relative_homology_group p X S"
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by (simp add: relative_homology_group_def)
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lemma nontrivial_relative_homology_group:
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fixes p::nat
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shows "relative_homology_group p X S
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= relcycle_group p X S Mod singular_relboundary_set p X S"
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by (simp add: relative_homology_group_def)
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lemma singular_relboundary_ss:
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"singular_relboundary p X S x \<Longrightarrow> Poly_Mapping.keys x \<subseteq> singular_simplex_set p X"
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using singular_chain_def singular_relboundary_imp_chain by blast
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lemma trivial_relative_homology_group [simp]:
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"p < 0 \<Longrightarrow> trivial_group(relative_homology_group p X S)"
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by (simp add: relative_homology_group_def)
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lemma subgroup_singular_relboundary:
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"subgroup (singular_relboundary_set p X S) (chain_group p X)"
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unfolding chain_group_def
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proof unfold_locales
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show "singular_relboundary_set p X S
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\<subseteq> carrier (free_Abelian_group (singular_simplex_set p X))"
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using singular_chain_def singular_relboundary_imp_chain by fastforce
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next
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fix x
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assume "x \<in> singular_relboundary_set p X S"
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then show "inv\<^bsub>free_Abelian_group (singular_simplex_set p X)\<^esub> x
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\<in> singular_relboundary_set p X S"
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by (simp add: singular_relboundary_ss singular_relboundary_minus)
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qed (auto simp: free_Abelian_group_def singular_relboundary_add)
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lemma subgroup_singular_relboundary_relcycle:
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"subgroup (singular_relboundary_set p X S) (relcycle_group p X S)"
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unfolding relcycle_group_def
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apply (rule group.subgroup_of_subgroup_generated)
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by (auto simp: singular_relcycle subgroup_singular_relboundary intro: singular_relboundary_imp_relcycle)
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lemma normal_subgroup_singular_relboundary_relcycle:
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"(singular_relboundary_set p X S) \<lhd> (relcycle_group p X S)"
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by (simp add: comm_group.normal_iff_subgroup subgroup_singular_relboundary_relcycle)
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lemma group_relative_homology_group [simp]:
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"group (relative_homology_group p X S)"
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by (simp add: relative_homology_group_def normal.factorgroup_is_group
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normal_subgroup_singular_relboundary_relcycle)
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lemma right_coset_singular_relboundary:
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"r_coset (relcycle_group p X S) (singular_relboundary_set p X S)
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= (\<lambda>a. {b. homologous_rel p X S a b})"
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using singular_relboundary_minus
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by (force simp: r_coset_def homologous_rel_def relcycle_group_def subgroup_generated_def)
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lemma carrier_relative_homology_group:
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"carrier(relative_homology_group (int p) X S)
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= (homologous_rel_set p X S) ` singular_relcycle_set p X S"
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by (auto simp: set_eq_iff image_iff relative_homology_group_def FactGroup_def RCOSETS_def right_coset_singular_relboundary)
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lemma carrier_relative_homology_group_0:
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"carrier(relative_homology_group 0 X S)
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= (homologous_rel_set 0 X S) ` singular_relcycle_set 0 X S"
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using carrier_relative_homology_group [of 0 X S] by simp
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lemma one_relative_homology_group [simp]:
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"one(relative_homology_group (int p) X S) = singular_relboundary_set p X S"
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by (simp add: relative_homology_group_def FactGroup_def)
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lemma mult_relative_homology_group:
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"(\<otimes>\<^bsub>relative_homology_group (int p) X S\<^esub>) = (\<lambda>R S. (\<Union>r\<in>R. \<Union>s\<in>S. {r + s}))"
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unfolding relcycle_group_def subgroup_generated_def chain_group_def free_Abelian_group_def set_mult_def relative_homology_group_def FactGroup_def
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by force
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lemma inv_relative_homology_group:
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assumes "R \<in> carrier (relative_homology_group (int p) X S)"
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shows "m_inv(relative_homology_group (int p) X S) R = uminus ` R"
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proof (rule group.inv_equality [OF group_relative_homology_group _ assms])
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obtain c where c: "R = homologous_rel_set p X S c" "singular_relcycle p X S c"
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using assms by (auto simp: carrier_relative_homology_group)
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have "singular_relboundary p X S (b - a)"
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if "a \<in> R" and "b \<in> R" for a b
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using c that
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by clarify (metis homologous_rel_def homologous_rel_eq)
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moreover
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have "x \<in> (\<Union>x\<in>R. \<Union>y\<in>R. {y - x})"
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if "singular_relboundary p X S x" for x
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using c
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by simp (metis diff_eq_eq homologous_rel_def homologous_rel_refl homologous_rel_sym that)
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ultimately
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have "(\<Union>x\<in>R. \<Union>xa\<in>R. {xa - x}) = singular_relboundary_set p X S"
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by auto
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then show "uminus ` R \<otimes>\<^bsub>relative_homology_group (int p) X S\<^esub> R =
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\<one>\<^bsub>relative_homology_group (int p) X S\<^esub>"
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by (auto simp: carrier_relative_homology_group mult_relative_homology_group)
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have "singular_relcycle p X S (-c)"
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using c by (simp add: singular_relcycle_minus)
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moreover have "homologous_rel p X S c x \<Longrightarrow> homologous_rel p X S (-c) (- x)" for x
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by (metis homologous_rel_def homologous_rel_sym minus_diff_eq minus_diff_minus)
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moreover have "homologous_rel p X S (-c) x \<Longrightarrow> x \<in> uminus ` homologous_rel_set p X S c" for x
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by (clarsimp simp: image_iff) (metis add.inverse_inverse diff_0 homologous_rel_diff homologous_rel_refl)
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ultimately show "uminus ` R \<in> carrier (relative_homology_group (int p) X S)"
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using c by (auto simp: carrier_relative_homology_group)
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qed
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lemma homologous_rel_eq_relboundary:
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"homologous_rel p X S c = singular_relboundary p X S
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\<longleftrightarrow> singular_relboundary p X S c" (is "?lhs = ?rhs")
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proof
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assume ?lhs
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then show ?rhs
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unfolding homologous_rel_def
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by (metis diff_zero singular_relboundary_0)
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next
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assume R: ?rhs
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show ?lhs
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unfolding homologous_rel_def
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using singular_relboundary_diff R by fastforce
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qed
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lemma homologous_rel_set_eq_relboundary:
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"homologous_rel_set p X S c = singular_relboundary_set p X S \<longleftrightarrow> singular_relboundary p X S c"
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by (auto simp flip: homologous_rel_eq_relboundary)
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text\<open>Lift the boundary and induced maps to homology groups. We totalize both
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quite aggressively to the appropriate group identity in all "undefined"
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situations, which makes several of the properties cleaner and simpler.\<close>
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lemma homomorphism_chain_boundary:
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"chain_boundary p \<in> hom (relcycle_group p X S) (relcycle_group(p - Suc 0) (subtopology X S) {})"
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(is "?h \<in> hom ?G ?H")
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proof (rule homI)
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show "\<And>x. x \<in> carrier ?G \<Longrightarrow> ?h x \<in> carrier ?H"
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by (auto simp: singular_relcycle_def mod_subset_def chain_boundary_boundary)
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qed (simp add: relcycle_group_def subgroup_generated_def chain_boundary_add)
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lemma hom_boundary1:
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"\<exists>d. \<forall>p X S.
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d p X S \<in> hom (relative_homology_group (int p) X S)
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(homology_group (int (p - Suc 0)) (subtopology X S))
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\<and> (\<forall>c. singular_relcycle p X S c
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\<longrightarrow> d p X S (homologous_rel_set p X S c)
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= homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c))"
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(is "\<exists>d. \<forall>p X S. ?\<Phi> (d p X S) p X S")
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proof ((subst choice_iff [symmetric])+, clarify)
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fix p X and S :: "'a set"
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define \<theta> where "\<theta> \<equiv> r_coset (relcycle_group(p - Suc 0) (subtopology X S) {})
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(singular_relboundary_set (p - Suc 0) (subtopology X S) {}) \<circ> chain_boundary p"
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define H where "H \<equiv> relative_homology_group (int (p - Suc 0)) (subtopology X S) {}"
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define J where "J \<equiv> relcycle_group (p - Suc 0) (subtopology X S) {}"
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have \<theta>: "\<theta> \<in> hom (relcycle_group p X S) H"
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unfolding \<theta>_def
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proof (rule hom_compose)
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show "chain_boundary p \<in> hom (relcycle_group p X S) J"
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by (simp add: J_def homomorphism_chain_boundary)
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show "(#>\<^bsub>relcycle_group (p - Suc 0) (subtopology X S) {}\<^esub>)
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(singular_relboundary_set (p - Suc 0) (subtopology X S) {}) \<in> hom J H"
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by (simp add: H_def J_def nontrivial_relative_homology_group
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normal.r_coset_hom_Mod normal_subgroup_singular_relboundary_relcycle)
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qed
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have *: "singular_relboundary (p - Suc 0) (subtopology X S) {} (chain_boundary p c)"
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if "singular_relboundary p X S c" for c
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proof (cases "p=0")
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case True
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then show ?thesis
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by (metis chain_boundary_def singular_relboundary_0)
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next
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case False
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with that have "\<exists>d. singular_chain p (subtopology X S) d \<and> chain_boundary p d = chain_boundary p c"
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by (metis add.left_neutral chain_boundary_add chain_boundary_boundary_alt singular_relboundary)
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with that False show ?thesis
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by (auto simp: singular_boundary)
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qed
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have \<theta>_eq: "\<theta> x = \<theta> y"
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if x: "x \<in> singular_relcycle_set p X S" and y: "y \<in> singular_relcycle_set p X S"
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and eq: "singular_relboundary_set p X S #>\<^bsub>relcycle_group p X S\<^esub> x
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= singular_relboundary_set p X S #>\<^bsub>relcycle_group p X S\<^esub> y" for x y
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proof -
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have "singular_relboundary p X S (x-y)"
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by (metis eq homologous_rel_def homologous_rel_eq mem_Collect_eq right_coset_singular_relboundary)
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with * have "(singular_relboundary (p - Suc 0) (subtopology X S) {}) (chain_boundary p (x-y))"
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by blast
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then show ?thesis
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unfolding \<theta>_def comp_def
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by (metis chain_boundary_diff homologous_rel_def homologous_rel_eq right_coset_singular_relboundary)
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qed
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obtain d
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where "d \<in> hom ((relcycle_group p X S) Mod (singular_relboundary_set p X S)) H"
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and d: "\<And>u. u \<in> singular_relcycle_set p X S \<Longrightarrow> d (homologous_rel_set p X S u) = \<theta> u"
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by (metis FactGroup_universal [OF \<theta> normal_subgroup_singular_relboundary_relcycle \<theta>_eq] right_coset_singular_relboundary carrier_relcycle_group)
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then have "d \<in> hom (relative_homology_group p X S) H"
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by (simp add: nontrivial_relative_homology_group)
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then show "\<exists>d. ?\<Phi> d p X S"
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by (force simp: H_def right_coset_singular_relboundary d \<theta>_def)
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qed
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lemma hom_boundary2:
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"\<exists>d. (\<forall>p X S.
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(d p X S) \<in> hom (relative_homology_group p X S)
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(homology_group (p - 1) (subtopology X S)))
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\<and> (\<forall>p X S c. singular_relcycle p X S c \<and> Suc 0 \<le> p
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\<longrightarrow> d p X S (homologous_rel_set p X S c)
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= homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c))"
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(is "\<exists>d. ?\<Phi> d")
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proof -
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have *: "\<exists>f. \<Phi>(\<lambda>p. if p \<le> 0 then \<lambda>q r t. undefined else f(nat p)) \<Longrightarrow> \<exists>f. \<Phi> f" for \<Phi>
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by blast
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show ?thesis
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apply (rule * [OF ex_forward [OF hom_boundary1]])
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apply (simp add: not_le relative_homology_group_def nat_diff_distrib' int_eq_iff nat_diff_distrib flip: nat_1)
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by (simp add: hom_def singleton_group_def)
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303 |
qed
|
|
304 |
|
|
305 |
lemma hom_boundary3:
|
|
306 |
"\<exists>d. ((\<forall>p X S c. c \<notin> carrier(relative_homology_group p X S)
|
|
307 |
\<longrightarrow> d p X S c = one(homology_group (p-1) (subtopology X S))) \<and>
|
|
308 |
(\<forall>p X S.
|
|
309 |
d p X S \<in> hom (relative_homology_group p X S)
|
|
310 |
(homology_group (p-1) (subtopology X S))) \<and>
|
|
311 |
(\<forall>p X S c.
|
|
312 |
singular_relcycle p X S c \<and> 1 \<le> p
|
|
313 |
\<longrightarrow> d p X S (homologous_rel_set p X S c)
|
|
314 |
= homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c)) \<and>
|
|
315 |
(\<forall>p X S. d p X S = d p X (topspace X \<inter> S))) \<and>
|
|
316 |
(\<forall>p X S c. d p X S c \<in> carrier(homology_group (p-1) (subtopology X S))) \<and>
|
|
317 |
(\<forall>p. p \<le> 0 \<longrightarrow> d p = (\<lambda>q r t. undefined))"
|
|
318 |
(is "\<exists>x. ?P x \<and> ?Q x \<and> ?R x")
|
|
319 |
proof -
|
|
320 |
have "\<And>x. ?Q x \<Longrightarrow> ?R x"
|
|
321 |
by (erule all_forward) (force simp: relative_homology_group_def)
|
|
322 |
moreover have "\<exists>x. ?P x \<and> ?Q x"
|
|
323 |
proof -
|
|
324 |
obtain d:: "[int, 'a topology, 'a set, ('a chain) set] \<Rightarrow> ('a chain) set"
|
|
325 |
where 1: "\<And>p X S. d p X S \<in> hom (relative_homology_group p X S)
|
|
326 |
(homology_group (p - 1) (subtopology X S))"
|
|
327 |
and 2: "\<And>n X S c. singular_relcycle n X S c \<and> Suc 0 \<le> n
|
|
328 |
\<Longrightarrow> d n X S (homologous_rel_set n X S c)
|
|
329 |
= homologous_rel_set (n - Suc 0) (subtopology X S) {} (chain_boundary n c)"
|
|
330 |
using hom_boundary2 by blast
|
|
331 |
have 4: "c \<in> carrier (relative_homology_group p X S) \<Longrightarrow>
|
|
332 |
d p X (topspace X \<inter> S) c \<in> carrier (relative_homology_group (p-1) (subtopology X S) {})"
|
|
333 |
for p X S c
|
|
334 |
using hom_carrier [OF 1 [of p X "topspace X \<inter> S"]]
|
|
335 |
by (simp add: image_subset_iff subtopology_restrict)
|
|
336 |
show ?thesis
|
|
337 |
apply (rule_tac x="\<lambda>p X S c.
|
|
338 |
if c \<in> carrier(relative_homology_group p X S)
|
|
339 |
then d p X (topspace X \<inter> S) c
|
|
340 |
else one(homology_group (p - 1) (subtopology X S))" in exI)
|
|
341 |
apply (simp add: Int_left_absorb subtopology_restrict carrier_relative_homology_group
|
|
342 |
group.is_monoid group.restrict_hom_iff 4 cong: if_cong)
|
|
343 |
apply (rule conjI)
|
|
344 |
apply (metis 1 relative_homology_group_restrict subtopology_restrict)
|
|
345 |
apply (metis 2 homologous_rel_restrict singular_relcycle_def subtopology_restrict)
|
|
346 |
done
|
|
347 |
qed
|
|
348 |
ultimately show ?thesis
|
|
349 |
by auto
|
|
350 |
qed
|
|
351 |
|
|
352 |
|
|
353 |
consts hom_boundary :: "[int,'a topology,'a set,'a chain set] \<Rightarrow> 'a chain set"
|
|
354 |
specification (hom_boundary)
|
|
355 |
hom_boundary:
|
|
356 |
"((\<forall>p X S c. c \<notin> carrier(relative_homology_group p X S)
|
|
357 |
\<longrightarrow> hom_boundary p X S c = one(homology_group (p-1) (subtopology X (S::'a set)))) \<and>
|
|
358 |
(\<forall>p X S.
|
|
359 |
hom_boundary p X S \<in> hom (relative_homology_group p X S)
|
|
360 |
(homology_group (p-1) (subtopology X (S::'a set)))) \<and>
|
|
361 |
(\<forall>p X S c.
|
|
362 |
singular_relcycle p X S c \<and> 1 \<le> p
|
|
363 |
\<longrightarrow> hom_boundary p X S (homologous_rel_set p X S c)
|
|
364 |
= homologous_rel_set (p - Suc 0) (subtopology X (S::'a set)) {} (chain_boundary p c)) \<and>
|
|
365 |
(\<forall>p X S. hom_boundary p X S = hom_boundary p X (topspace X \<inter> (S::'a set)))) \<and>
|
|
366 |
(\<forall>p X S c. hom_boundary p X S c \<in> carrier(homology_group (p-1) (subtopology X (S::'a set)))) \<and>
|
|
367 |
(\<forall>p. p \<le> 0 \<longrightarrow> hom_boundary p = (\<lambda>q r. \<lambda>t::'a chain set. undefined))"
|
|
368 |
by (fact hom_boundary3)
|
|
369 |
|
|
370 |
lemma hom_boundary_default:
|
|
371 |
"c \<notin> carrier(relative_homology_group p X S)
|
|
372 |
\<Longrightarrow> hom_boundary p X S c = one(homology_group (p-1) (subtopology X S))"
|
|
373 |
and hom_boundary_hom: "hom_boundary p X S \<in> hom (relative_homology_group p X S) (homology_group (p-1) (subtopology X S))"
|
|
374 |
and hom_boundary_restrict [simp]: "hom_boundary p X (topspace X \<inter> S) = hom_boundary p X S"
|
|
375 |
and hom_boundary_carrier: "hom_boundary p X S c \<in> carrier(homology_group (p-1) (subtopology X S))"
|
|
376 |
and hom_boundary_trivial: "p \<le> 0 \<Longrightarrow> hom_boundary p = (\<lambda>q r t. undefined)"
|
|
377 |
by (metis hom_boundary)+
|
|
378 |
|
|
379 |
lemma hom_boundary_chain_boundary:
|
|
380 |
"\<lbrakk>singular_relcycle p X S c; 1 \<le> p\<rbrakk>
|
|
381 |
\<Longrightarrow> hom_boundary (int p) X S (homologous_rel_set p X S c) =
|
|
382 |
homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c)"
|
|
383 |
by (metis hom_boundary)+
|
|
384 |
|
|
385 |
lemma hom_chain_map:
|
|
386 |
"\<lbrakk>continuous_map X Y f; f ` S \<subseteq> T\<rbrakk>
|
|
387 |
\<Longrightarrow> (chain_map p f) \<in> hom (relcycle_group p X S) (relcycle_group p Y T)"
|
|
388 |
by (force simp: chain_map_add singular_relcycle_chain_map hom_def)
|
|
389 |
|
|
390 |
|
|
391 |
lemma hom_induced1:
|
|
392 |
"\<exists>hom_relmap.
|
|
393 |
(\<forall>p X S Y T f.
|
|
394 |
continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T
|
|
395 |
\<longrightarrow> (hom_relmap p X S Y T f) \<in> hom (relative_homology_group (int p) X S)
|
|
396 |
(relative_homology_group (int p) Y T)) \<and>
|
|
397 |
(\<forall>p X S Y T f c.
|
|
398 |
continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
399 |
singular_relcycle p X S c
|
|
400 |
\<longrightarrow> hom_relmap p X S Y T f (homologous_rel_set p X S c) =
|
|
401 |
homologous_rel_set p Y T (chain_map p f c))"
|
|
402 |
proof -
|
|
403 |
have "\<exists>y. (y \<in> hom (relative_homology_group (int p) X S) (relative_homology_group (int p) Y T)) \<and>
|
|
404 |
(\<forall>c. singular_relcycle p X S c \<longrightarrow>
|
|
405 |
y (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c))"
|
|
406 |
if contf: "continuous_map X Y f" and fim: "f ` (topspace X \<inter> S) \<subseteq> T"
|
|
407 |
for p X S Y T and f :: "'a \<Rightarrow> 'b"
|
|
408 |
proof -
|
|
409 |
let ?f = "(#>\<^bsub>relcycle_group p Y T\<^esub>) (singular_relboundary_set p Y T) \<circ> chain_map p f"
|
|
410 |
let ?F = "\<lambda>x. singular_relboundary_set p X S #>\<^bsub>relcycle_group p X S\<^esub> x"
|
|
411 |
have 1: "?f \<in> hom (relcycle_group p X S) (relative_homology_group (int p) Y T)"
|
|
412 |
apply (rule hom_compose [where H = "relcycle_group p Y T"])
|
|
413 |
apply (metis contf fim hom_chain_map relcycle_group_restrict)
|
|
414 |
by (simp add: nontrivial_relative_homology_group normal.r_coset_hom_Mod normal_subgroup_singular_relboundary_relcycle)
|
|
415 |
have 2: "singular_relboundary_set p X S \<lhd> relcycle_group p X S"
|
|
416 |
using normal_subgroup_singular_relboundary_relcycle by blast
|
|
417 |
have 3: "?f x = ?f y"
|
|
418 |
if "singular_relcycle p X S x" "singular_relcycle p X S y" "?F x = ?F y" for x y
|
|
419 |
proof -
|
|
420 |
have "singular_relboundary p Y T (chain_map p f (x - y))"
|
|
421 |
apply (rule singular_relboundary_chain_map [OF _ contf fim])
|
|
422 |
by (metis homologous_rel_def homologous_rel_eq mem_Collect_eq right_coset_singular_relboundary singular_relboundary_restrict that(3))
|
|
423 |
then have "singular_relboundary p Y T (chain_map p f x - chain_map p f y)"
|
|
424 |
by (simp add: chain_map_diff)
|
|
425 |
with that
|
|
426 |
show ?thesis
|
|
427 |
apply (simp add: right_coset_singular_relboundary homologous_rel_set_eq)
|
|
428 |
apply (simp add: homologous_rel_def)
|
|
429 |
done
|
|
430 |
qed
|
|
431 |
obtain g where "g \<in> hom (relcycle_group p X S Mod singular_relboundary_set p X S)
|
|
432 |
(relative_homology_group (int p) Y T)"
|
|
433 |
"\<And>x. x \<in> singular_relcycle_set p X S \<Longrightarrow> g (?F x) = ?f x"
|
|
434 |
using FactGroup_universal [OF 1 2 3, unfolded carrier_relcycle_group] by blast
|
|
435 |
then show ?thesis
|
|
436 |
by (force simp: right_coset_singular_relboundary nontrivial_relative_homology_group)
|
|
437 |
qed
|
|
438 |
then show ?thesis
|
|
439 |
apply (simp flip: all_conj_distrib)
|
|
440 |
apply ((subst choice_iff [symmetric])+)
|
|
441 |
apply metis
|
|
442 |
done
|
|
443 |
qed
|
|
444 |
|
|
445 |
lemma hom_induced2:
|
|
446 |
"\<exists>hom_relmap.
|
|
447 |
(\<forall>p X S Y T f.
|
|
448 |
continuous_map X Y f \<and>
|
|
449 |
f ` (topspace X \<inter> S) \<subseteq> T
|
|
450 |
\<longrightarrow> (hom_relmap p X S Y T f) \<in> hom (relative_homology_group p X S)
|
|
451 |
(relative_homology_group p Y T)) \<and>
|
|
452 |
(\<forall>p X S Y T f c.
|
|
453 |
continuous_map X Y f \<and>
|
|
454 |
f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
455 |
singular_relcycle p X S c
|
|
456 |
\<longrightarrow> hom_relmap p X S Y T f (homologous_rel_set p X S c) =
|
|
457 |
homologous_rel_set p Y T (chain_map p f c)) \<and>
|
|
458 |
(\<forall>p. p < 0 \<longrightarrow> hom_relmap p = (\<lambda>X S Y T f c. undefined))"
|
|
459 |
(is "\<exists>d. ?\<Phi> d")
|
|
460 |
proof -
|
|
461 |
have *: "\<exists>f. \<Phi>(\<lambda>p. if p < 0 then \<lambda>X S Y T f c. undefined else f(nat p)) \<Longrightarrow> \<exists>f. \<Phi> f" for \<Phi>
|
|
462 |
by blast
|
|
463 |
show ?thesis
|
|
464 |
apply (rule * [OF ex_forward [OF hom_induced1]])
|
|
465 |
apply (simp add: not_le relative_homology_group_def nat_diff_distrib' int_eq_iff nat_diff_distrib flip: nat_1)
|
|
466 |
done
|
|
467 |
qed
|
|
468 |
|
|
469 |
lemma hom_induced3:
|
|
470 |
"\<exists>hom_relmap.
|
|
471 |
((\<forall>p X S Y T f c.
|
|
472 |
~(continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
473 |
c \<in> carrier(relative_homology_group p X S))
|
|
474 |
\<longrightarrow> hom_relmap p X S Y T f c = one(relative_homology_group p Y T)) \<and>
|
|
475 |
(\<forall>p X S Y T f.
|
|
476 |
hom_relmap p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)) \<and>
|
|
477 |
(\<forall>p X S Y T f c.
|
|
478 |
continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> singular_relcycle p X S c
|
|
479 |
\<longrightarrow> hom_relmap p X S Y T f (homologous_rel_set p X S c) =
|
|
480 |
homologous_rel_set p Y T (chain_map p f c)) \<and>
|
|
481 |
(\<forall>p X S Y T.
|
|
482 |
hom_relmap p X S Y T =
|
|
483 |
hom_relmap p X (topspace X \<inter> S) Y (topspace Y \<inter> T))) \<and>
|
|
484 |
(\<forall>p X S Y f T c.
|
|
485 |
hom_relmap p X S Y T f c \<in> carrier(relative_homology_group p Y T)) \<and>
|
|
486 |
(\<forall>p. p < 0 \<longrightarrow> hom_relmap p = (\<lambda>X S Y T f c. undefined))"
|
|
487 |
(is "\<exists>x. ?P x \<and> ?Q x \<and> ?R x")
|
|
488 |
proof -
|
|
489 |
have "\<And>x. ?Q x \<Longrightarrow> ?R x"
|
|
490 |
by (erule all_forward) (fastforce simp: relative_homology_group_def)
|
|
491 |
moreover have "\<exists>x. ?P x \<and> ?Q x"
|
|
492 |
proof -
|
|
493 |
obtain hom_relmap:: "[int,'a topology,'a set,'b topology,'b set,'a \<Rightarrow> 'b,('a chain) set] \<Rightarrow> ('b chain) set"
|
|
494 |
where 1: "\<And>p X S Y T f. \<lbrakk>continuous_map X Y f; f ` (topspace X \<inter> S) \<subseteq> T\<rbrakk> \<Longrightarrow>
|
|
495 |
hom_relmap p X S Y T f
|
|
496 |
\<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)"
|
|
497 |
and 2: "\<And>p X S Y T f c.
|
|
498 |
\<lbrakk>continuous_map X Y f; f ` (topspace X \<inter> S) \<subseteq> T; singular_relcycle p X S c\<rbrakk>
|
|
499 |
\<Longrightarrow>
|
|
500 |
hom_relmap (int p) X S Y T f (homologous_rel_set p X S c) =
|
|
501 |
homologous_rel_set p Y T (chain_map p f c)"
|
|
502 |
and 3: "(\<forall>p. p < 0 \<longrightarrow> hom_relmap p = (\<lambda>X S Y T f c. undefined))"
|
|
503 |
using hom_induced2 [where ?'a='a and ?'b='b]
|
|
504 |
apply clarify
|
|
505 |
apply (rule_tac hom_relmap=hom_relmap in that, auto)
|
|
506 |
done
|
|
507 |
have 4: "\<lbrakk>continuous_map X Y f; f ` (topspace X \<inter> S) \<subseteq> T; c \<in> carrier (relative_homology_group p X S)\<rbrakk> \<Longrightarrow>
|
|
508 |
hom_relmap p X (topspace X \<inter> S) Y (topspace Y \<inter> T) f c
|
|
509 |
\<in> carrier (relative_homology_group p Y T)"
|
|
510 |
for p X S Y f T c
|
|
511 |
using hom_carrier [OF 1 [of X Y f "topspace X \<inter> S" "topspace Y \<inter> T" p]]
|
|
512 |
by (simp add: image_subset_iff subtopology_restrict continuous_map_def)
|
|
513 |
have inhom: "(\<lambda>c. if continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
514 |
c \<in> carrier (relative_homology_group p X S)
|
|
515 |
then hom_relmap p X (topspace X \<inter> S) Y (topspace Y \<inter> T) f c
|
|
516 |
else \<one>\<^bsub>relative_homology_group p Y T\<^esub>)
|
|
517 |
\<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)" (is "?h \<in> hom ?GX ?GY")
|
|
518 |
for p X S Y T f
|
|
519 |
proof (rule homI)
|
|
520 |
show "\<And>x. x \<in> carrier ?GX \<Longrightarrow> ?h x \<in> carrier ?GY"
|
|
521 |
by (auto simp: 4 group.is_monoid)
|
|
522 |
show "?h (x \<otimes>\<^bsub>?GX\<^esub> y) = ?h x \<otimes>\<^bsub>?GY\<^esub>?h y" if "x \<in> carrier ?GX" "y \<in> carrier ?GX" for x y
|
|
523 |
proof (cases "p < 0")
|
|
524 |
case True
|
|
525 |
with that show ?thesis
|
|
526 |
by (simp add: relative_homology_group_def singleton_group_def 3)
|
|
527 |
next
|
|
528 |
case False
|
|
529 |
show ?thesis
|
|
530 |
proof (cases "continuous_map X Y f")
|
|
531 |
case True
|
|
532 |
then have "f ` (topspace X \<inter> S) \<subseteq> topspace Y"
|
|
533 |
by (meson IntE continuous_map_def image_subsetI)
|
|
534 |
then show ?thesis
|
|
535 |
using True False that
|
|
536 |
using 1 [of X Y f "topspace X \<inter> S" "topspace Y \<inter> T" p]
|
|
537 |
by (simp add: 4 continuous_map_image_subset_topspace hom_mult not_less group.is_monoid monoid.m_closed Int_left_absorb)
|
|
538 |
qed (simp add: group.is_monoid)
|
|
539 |
qed
|
|
540 |
qed
|
|
541 |
have hrel: "\<lbrakk>continuous_map X Y f; f ` (topspace X \<inter> S) \<subseteq> T; singular_relcycle p X S c\<rbrakk>
|
|
542 |
\<Longrightarrow> hom_relmap (int p) X (topspace X \<inter> S) Y (topspace Y \<inter> T)
|
|
543 |
f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)"
|
|
544 |
for p X S Y T f c
|
|
545 |
using 2 [of X Y f "topspace X \<inter> S" "topspace Y \<inter> T" p c]
|
|
546 |
by simp (meson IntE continuous_map_def image_subsetI)
|
|
547 |
show ?thesis
|
|
548 |
apply (rule_tac x="\<lambda>p X S Y T f c.
|
|
549 |
if continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
550 |
c \<in> carrier(relative_homology_group p X S)
|
|
551 |
then hom_relmap p X (topspace X \<inter> S) Y (topspace Y \<inter> T) f c
|
|
552 |
else one(relative_homology_group p Y T)" in exI)
|
|
553 |
apply (simp add: Int_left_absorb subtopology_restrict carrier_relative_homology_group
|
|
554 |
group.is_monoid group.restrict_hom_iff 4 inhom hrel cong: if_cong)
|
|
555 |
apply (force simp: continuous_map_def intro!: ext)
|
|
556 |
done
|
|
557 |
qed
|
|
558 |
ultimately show ?thesis
|
|
559 |
by auto
|
|
560 |
qed
|
|
561 |
|
|
562 |
|
|
563 |
consts hom_induced:: "[int,'a topology,'a set,'b topology,'b set,'a \<Rightarrow> 'b,('a chain) set] \<Rightarrow> ('b chain) set"
|
|
564 |
specification (hom_induced)
|
|
565 |
hom_induced:
|
|
566 |
"((\<forall>p X S Y T f c.
|
|
567 |
~(continuous_map X Y f \<and>
|
|
568 |
f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
569 |
c \<in> carrier(relative_homology_group p X S))
|
|
570 |
\<longrightarrow> hom_induced p X (S::'a set) Y (T::'b set) f c =
|
|
571 |
one(relative_homology_group p Y T)) \<and>
|
|
572 |
(\<forall>p X S Y T f.
|
|
573 |
(hom_induced p X (S::'a set) Y (T::'b set) f) \<in> hom (relative_homology_group p X S)
|
|
574 |
(relative_homology_group p Y T)) \<and>
|
|
575 |
(\<forall>p X S Y T f c.
|
|
576 |
continuous_map X Y f \<and>
|
|
577 |
f ` (topspace X \<inter> S) \<subseteq> T \<and>
|
|
578 |
singular_relcycle p X S c
|
|
579 |
\<longrightarrow> hom_induced p X (S::'a set) Y (T::'b set) f (homologous_rel_set p X S c) =
|
|
580 |
homologous_rel_set p Y T (chain_map p f c)) \<and>
|
|
581 |
(\<forall>p X S Y T.
|
|
582 |
hom_induced p X (S::'a set) Y (T::'b set) =
|
|
583 |
hom_induced p X (topspace X \<inter> S) Y (topspace Y \<inter> T))) \<and>
|
|
584 |
(\<forall>p X S Y f T c.
|
|
585 |
hom_induced p X (S::'a set) Y (T::'b set) f c \<in>
|
|
586 |
carrier(relative_homology_group p Y T)) \<and>
|
|
587 |
(\<forall>p. p < 0 \<longrightarrow> hom_induced p = (\<lambda>X S Y T. \<lambda>f::'a\<Rightarrow>'b. \<lambda>c. undefined))"
|
|
588 |
by (fact hom_induced3)
|
|
589 |
|
|
590 |
lemma hom_induced_default:
|
|
591 |
"~(continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> c \<in> carrier(relative_homology_group p X S))
|
|
592 |
\<Longrightarrow> hom_induced p X S Y T f c = one(relative_homology_group p Y T)"
|
|
593 |
and hom_induced_hom:
|
|
594 |
"hom_induced p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)"
|
|
595 |
and hom_induced_restrict [simp]:
|
|
596 |
"hom_induced p X (topspace X \<inter> S) Y (topspace Y \<inter> T) = hom_induced p X S Y T"
|
|
597 |
and hom_induced_carrier:
|
|
598 |
"hom_induced p X S Y T f c \<in> carrier(relative_homology_group p Y T)"
|
|
599 |
and hom_induced_trivial: "p < 0 \<Longrightarrow> hom_induced p = (\<lambda>X S Y T f c. undefined)"
|
|
600 |
by (metis hom_induced)+
|
|
601 |
|
|
602 |
lemma hom_induced_chain_map_gen:
|
|
603 |
"\<lbrakk>continuous_map X Y f; f ` (topspace X \<inter> S) \<subseteq> T; singular_relcycle p X S c\<rbrakk>
|
|
604 |
\<Longrightarrow> hom_induced p X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)"
|
|
605 |
by (metis hom_induced)
|
|
606 |
|
|
607 |
lemma hom_induced_chain_map:
|
|
608 |
"\<lbrakk>continuous_map X Y f; f ` S \<subseteq> T; singular_relcycle p X S c\<rbrakk>
|
|
609 |
\<Longrightarrow> hom_induced p X S Y T f (homologous_rel_set p X S c)
|
|
610 |
= homologous_rel_set p Y T (chain_map p f c)"
|
|
611 |
by (meson Int_lower2 hom_induced image_subsetI image_subset_iff subset_iff)
|
|
612 |
|
|
613 |
|
|
614 |
lemma hom_induced_eq:
|
|
615 |
assumes "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x"
|
|
616 |
shows "hom_induced p X S Y T f = hom_induced p X S Y T g"
|
|
617 |
proof -
|
|
618 |
consider "p < 0" | n where "p = int n"
|
|
619 |
by (metis int_nat_eq not_less)
|
|
620 |
then show ?thesis
|
|
621 |
proof cases
|
|
622 |
case 1
|
|
623 |
then show ?thesis
|
|
624 |
by (simp add: hom_induced_trivial)
|
|
625 |
next
|
|
626 |
case 2
|
|
627 |
have "hom_induced n X S Y T f C = hom_induced n X S Y T g C" for C
|
|
628 |
proof -
|
|
629 |
have "continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> C \<in> carrier (relative_homology_group n X S)
|
|
630 |
\<longleftrightarrow> continuous_map X Y g \<and> g ` (topspace X \<inter> S) \<subseteq> T \<and> C \<in> carrier (relative_homology_group n X S)"
|
|
631 |
(is "?P = ?Q")
|
|
632 |
by (metis IntD1 assms continuous_map_eq image_cong)
|
|
633 |
then consider "\<not> ?P \<and> \<not> ?Q" | "?P \<and> ?Q"
|
|
634 |
by blast
|
|
635 |
then show ?thesis
|
|
636 |
proof cases
|
|
637 |
case 1
|
|
638 |
then show ?thesis
|
|
639 |
by (simp add: hom_induced_default)
|
|
640 |
next
|
|
641 |
case 2
|
|
642 |
have "homologous_rel_set n Y T (chain_map n f c) = homologous_rel_set n Y T (chain_map n g c)"
|
|
643 |
if "continuous_map X Y f" "f ` (topspace X \<inter> S) \<subseteq> T"
|
|
644 |
"continuous_map X Y g" "g ` (topspace X \<inter> S) \<subseteq> T"
|
|
645 |
"C = homologous_rel_set n X S c" "singular_relcycle n X S c"
|
|
646 |
for c
|
|
647 |
proof -
|
|
648 |
have "chain_map n f c = chain_map n g c"
|
|
649 |
using assms chain_map_eq singular_relcycle that by blast
|
|
650 |
then show ?thesis
|
|
651 |
by simp
|
|
652 |
qed
|
|
653 |
with 2 show ?thesis
|
|
654 |
by (auto simp: relative_homology_group_def carrier_FactGroup
|
|
655 |
right_coset_singular_relboundary hom_induced_chain_map_gen)
|
|
656 |
qed
|
|
657 |
qed
|
|
658 |
with 2 show ?thesis
|
|
659 |
by auto
|
|
660 |
qed
|
|
661 |
qed
|
|
662 |
|
|
663 |
|
|
664 |
subsection\<open>Towards the Eilenberg-Steenrod axioms\<close>
|
|
665 |
|
|
666 |
text\<open>First prove we get functors into abelian groups with the boundary map
|
|
667 |
being a natural transformation between them, and prove Eilenberg-Steenrod
|
|
668 |
axioms (we also prove additivity a bit later on if one counts that). \<close>
|
|
669 |
(*1. Exact sequence from the inclusions and boundary map
|
|
670 |
H_{p+1} X --(j')\<longlongrightarrow> H_{p+1}X (A) --(d')\<longlongrightarrow> H_p A --(i')\<longlongrightarrow> H_p X
|
|
671 |
2. Dimension axiom: H_p X is trivial for one-point X and p =/= 0
|
|
672 |
3. Homotopy invariance of the induced map
|
|
673 |
4. Excision: inclusion (X - U,A - U) --(i')\<longlongrightarrow> X (A) induces an isomorphism
|
|
674 |
when cl U \<subseteq> int A*)
|
|
675 |
|
|
676 |
|
|
677 |
lemma abelian_relative_homology_group [simp]:
|
|
678 |
"comm_group(relative_homology_group p X S)"
|
|
679 |
apply (simp add: relative_homology_group_def)
|
|
680 |
apply (metis comm_group.abelian_FactGroup abelian_relcycle_group subgroup_singular_relboundary_relcycle)
|
|
681 |
done
|
|
682 |
|
|
683 |
lemma abelian_homology_group: "comm_group(homology_group p X)"
|
|
684 |
by simp
|
|
685 |
|
|
686 |
|
|
687 |
lemma hom_induced_id_gen:
|
|
688 |
assumes contf: "continuous_map X X f" and feq: "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x"
|
|
689 |
and c: "c \<in> carrier (relative_homology_group p X S)"
|
|
690 |
shows "hom_induced p X S X S f c = c"
|
|
691 |
proof -
|
|
692 |
consider "p < 0" | n where "p = int n"
|
|
693 |
by (metis int_nat_eq not_less)
|
|
694 |
then show ?thesis
|
|
695 |
proof cases
|
|
696 |
case 1
|
|
697 |
with c show ?thesis
|
|
698 |
by (simp add: hom_induced_trivial relative_homology_group_def)
|
|
699 |
next
|
|
700 |
case 2
|
|
701 |
have cm: "chain_map n f d = d" if "singular_relcycle n X S d" for d
|
|
702 |
using that assms by (auto simp: chain_map_id_gen singular_relcycle)
|
|
703 |
have "f ` (topspace X \<inter> S) \<subseteq> S"
|
|
704 |
using feq by auto
|
|
705 |
with 2 c show ?thesis
|
|
706 |
by (auto simp: nontrivial_relative_homology_group carrier_FactGroup
|
|
707 |
cm right_coset_singular_relboundary hom_induced_chain_map_gen assms)
|
|
708 |
qed
|
|
709 |
qed
|
|
710 |
|
|
711 |
|
|
712 |
lemma hom_induced_id:
|
|
713 |
"c \<in> carrier (relative_homology_group p X S) \<Longrightarrow> hom_induced p X S X S id c = c"
|
|
714 |
by (rule hom_induced_id_gen) auto
|
|
715 |
|
|
716 |
lemma hom_induced_compose:
|
|
717 |
assumes "continuous_map X Y f" "f ` S \<subseteq> T" "continuous_map Y Z g" "g ` T \<subseteq> U"
|
|
718 |
shows "hom_induced p X S Z U (g \<circ> f) = hom_induced p Y T Z U g \<circ> hom_induced p X S Y T f"
|
|
719 |
proof -
|
|
720 |
consider (neg) "p < 0" | (int) n where "p = int n"
|
|
721 |
by (metis int_nat_eq not_less)
|
|
722 |
then show ?thesis
|
|
723 |
proof cases
|
|
724 |
case int
|
|
725 |
have gf: "continuous_map X Z (g \<circ> f)"
|
|
726 |
using assms continuous_map_compose by fastforce
|
|
727 |
have gfim: "(g \<circ> f) ` S \<subseteq> U"
|
|
728 |
unfolding o_def using assms by blast
|
|
729 |
have sr: "\<And>a. singular_relcycle n X S a \<Longrightarrow> singular_relcycle n Y T (chain_map n f a)"
|
|
730 |
by (simp add: assms singular_relcycle_chain_map)
|
|
731 |
show ?thesis
|
|
732 |
proof
|
|
733 |
fix c
|
|
734 |
show "hom_induced p X S Z U (g \<circ> f) c = (hom_induced p Y T Z U g \<circ> hom_induced p X S Y T f) c"
|
|
735 |
proof (cases "c \<in> carrier(relative_homology_group p X S)")
|
|
736 |
case True
|
|
737 |
with gfim show ?thesis
|
|
738 |
unfolding int
|
|
739 |
by (auto simp: carrier_relative_homology_group gf gfim assms sr chain_map_compose hom_induced_chain_map)
|
|
740 |
next
|
|
741 |
case False
|
|
742 |
then show ?thesis
|
|
743 |
by (simp add: hom_induced_default hom_one [OF hom_induced_hom])
|
|
744 |
qed
|
|
745 |
qed
|
|
746 |
qed (force simp: hom_induced_trivial)
|
|
747 |
qed
|
|
748 |
|
|
749 |
lemma hom_induced_compose':
|
|
750 |
assumes "continuous_map X Y f" "f ` S \<subseteq> T" "continuous_map Y Z g" "g ` T \<subseteq> U"
|
|
751 |
shows "hom_induced p Y T Z U g (hom_induced p X S Y T f x) = hom_induced p X S Z U (g \<circ> f) x"
|
|
752 |
using hom_induced_compose [OF assms] by simp
|
|
753 |
|
|
754 |
lemma naturality_hom_induced:
|
|
755 |
assumes "continuous_map X Y f" "f ` S \<subseteq> T"
|
|
756 |
shows "hom_boundary q Y T \<circ> hom_induced q X S Y T f
|
|
757 |
= hom_induced (q - 1) (subtopology X S) {} (subtopology Y T) {} f \<circ> hom_boundary q X S"
|
|
758 |
proof (cases "q \<le> 0")
|
|
759 |
case False
|
|
760 |
then obtain p where p1: "p \<ge> Suc 0" and q: "q = int p"
|
|
761 |
using zero_le_imp_eq_int by force
|
|
762 |
show ?thesis
|
|
763 |
proof
|
|
764 |
fix c
|
|
765 |
show "(hom_boundary q Y T \<circ> hom_induced q X S Y T f) c =
|
|
766 |
(hom_induced (q - 1) (subtopology X S) {} (subtopology Y T) {} f \<circ> hom_boundary q X S) c"
|
|
767 |
proof (cases "c \<in> carrier(relative_homology_group p X S)")
|
|
768 |
case True
|
|
769 |
then obtain a where ceq: "c = homologous_rel_set p X S a" and a: "singular_relcycle p X S a"
|
|
770 |
by (force simp: carrier_relative_homology_group)
|
|
771 |
then have sr: "singular_relcycle p Y T (chain_map p f a)"
|
|
772 |
using assms singular_relcycle_chain_map by fastforce
|
|
773 |
then have sb: "singular_relcycle (p - Suc 0) (subtopology X S) {} (chain_boundary p a)"
|
|
774 |
by (metis One_nat_def a chain_boundary_boundary singular_chain_0 singular_relcycle)
|
|
775 |
have p1_eq: "int p - 1 = int (p - Suc 0)"
|
|
776 |
using p1 by auto
|
|
777 |
have cbm: "(chain_boundary p (chain_map p f a))
|
|
778 |
= (chain_map (p - Suc 0) f (chain_boundary p a))"
|
|
779 |
using a chain_boundary_chain_map singular_relcycle by blast
|
|
780 |
have contf: "continuous_map (subtopology X S) (subtopology Y T) f"
|
|
781 |
using assms
|
|
782 |
by (auto simp: continuous_map_in_subtopology topspace_subtopology
|
|
783 |
continuous_map_from_subtopology)
|
|
784 |
show ?thesis
|
|
785 |
unfolding q using assms p1 a
|
|
786 |
apply (simp add: ceq assms hom_induced_chain_map hom_boundary_chain_boundary
|
|
787 |
hom_boundary_chain_boundary [OF sr] singular_relcycle_def mod_subset_def)
|
|
788 |
apply (simp add: p1_eq contf sb cbm hom_induced_chain_map)
|
|
789 |
done
|
|
790 |
next
|
|
791 |
case False
|
|
792 |
with assms show ?thesis
|
|
793 |
unfolding q o_def using assms
|
|
794 |
apply (simp add: hom_induced_default hom_boundary_default)
|
|
795 |
by (metis group_relative_homology_group hom_boundary hom_induced hom_one one_relative_homology_group)
|
|
796 |
qed
|
|
797 |
qed
|
|
798 |
qed (force simp: hom_induced_trivial hom_boundary_trivial)
|
|
799 |
|
|
800 |
|
|
801 |
|
|
802 |
lemma homology_exactness_axiom_1:
|
|
803 |
"exact_seq ([homology_group (p-1) (subtopology X S), relative_homology_group p X S, homology_group p X],
|
|
804 |
[hom_boundary p X S,hom_induced p X {} X S id])"
|
|
805 |
proof -
|
|
806 |
consider (neg) "p < 0" | (int) n where "p = int n"
|
|
807 |
by (metis int_nat_eq not_less)
|
|
808 |
then have "(hom_induced p X {} X S id) ` carrier (homology_group p X)
|
|
809 |
= kernel (relative_homology_group p X S) (homology_group (p-1) (subtopology X S))
|
|
810 |
(hom_boundary p X S)"
|
|
811 |
proof cases
|
|
812 |
case neg
|
|
813 |
then show ?thesis
|
|
814 |
unfolding kernel_def singleton_group_def relative_homology_group_def
|
|
815 |
by (auto simp: hom_induced_trivial hom_boundary_trivial)
|
|
816 |
next
|
|
817 |
case int
|
|
818 |
have "hom_induced (int m) X {} X S id ` carrier (relative_homology_group (int m) X {})
|
|
819 |
= carrier (relative_homology_group (int m) X S) \<inter>
|
|
820 |
{c. hom_boundary (int m) X S c = \<one>\<^bsub>relative_homology_group (int m - 1) (subtopology X S) {}\<^esub>}" for m
|
|
821 |
proof (cases m)
|
|
822 |
case 0
|
|
823 |
have "hom_induced 0 X {} X S id ` carrier (relative_homology_group 0 X {})
|
|
824 |
= carrier (relative_homology_group 0 X S)" (is "?lhs = ?rhs")
|
|
825 |
proof
|
|
826 |
show "?lhs \<subseteq> ?rhs"
|
|
827 |
using hom_induced_hom [of 0 X "{}" X S id]
|
|
828 |
by (simp add: hom_induced_hom hom_carrier)
|
|
829 |
show "?rhs \<subseteq> ?lhs"
|
|
830 |
apply (clarsimp simp add: image_iff carrier_relative_homology_group [of 0, simplified] singular_relcycle)
|
|
831 |
apply (force simp: chain_map_id_gen chain_boundary_def singular_relcycle
|
|
832 |
hom_induced_chain_map [of concl: 0, simplified])
|
|
833 |
done
|
|
834 |
qed
|
|
835 |
with 0 show ?thesis
|
|
836 |
by (simp add: hom_boundary_trivial relative_homology_group_def [of "-1"] singleton_group_def)
|
|
837 |
next
|
|
838 |
case (Suc n)
|
|
839 |
have "(hom_induced (int (Suc n)) X {} X S id \<circ>
|
|
840 |
homologous_rel_set (Suc n) X {}) ` singular_relcycle_set (Suc n) X {}
|
|
841 |
= homologous_rel_set (Suc n) X S `
|
|
842 |
(singular_relcycle_set (Suc n) X S \<inter>
|
|
843 |
{c. hom_boundary (int (Suc n)) X S (homologous_rel_set (Suc n) X S c)
|
|
844 |
= singular_relboundary_set n (subtopology X S) {}})"
|
|
845 |
(is "?lhs = ?rhs")
|
|
846 |
proof -
|
|
847 |
have 1: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<longleftrightarrow> x \<in> C) \<Longrightarrow> f ` (A \<inter> B) = f ` (A \<inter> C)" for f A B C
|
|
848 |
by blast
|
|
849 |
have 2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> f x = f y; \<And>x. x \<in> B \<Longrightarrow> \<exists>y. y \<in> A \<and> f x = f y\<rbrakk>
|
|
850 |
\<Longrightarrow> f ` A = f ` B" for f A B
|
|
851 |
by blast
|
|
852 |
have "?lhs = homologous_rel_set (Suc n) X S ` singular_relcycle_set (Suc n) X {}"
|
|
853 |
apply (rule image_cong [OF refl])
|
|
854 |
apply (simp add: o_def hom_induced_chain_map chain_map_ident [of _ X] singular_relcycle
|
|
855 |
del: of_nat_Suc)
|
|
856 |
done
|
|
857 |
also have "\<dots> = homologous_rel_set (Suc n) X S `
|
|
858 |
(singular_relcycle_set (Suc n) X S \<inter>
|
|
859 |
{c. singular_relboundary n (subtopology X S) {} (chain_boundary (Suc n) c)})"
|
|
860 |
proof (rule 2)
|
|
861 |
fix c
|
|
862 |
assume "c \<in> singular_relcycle_set (Suc n) X {}"
|
|
863 |
then show "\<exists>y. y \<in> singular_relcycle_set (Suc n) X S \<inter>
|
|
864 |
{c. singular_relboundary n (subtopology X S) {} (chain_boundary (Suc n) c)} \<and>
|
|
865 |
homologous_rel_set (Suc n) X S c = homologous_rel_set (Suc n) X S y"
|
|
866 |
apply (rule_tac x=c in exI)
|
|
867 |
by (simp add: singular_boundary) (metis chain_boundary_0 singular_cycle singular_relcycle singular_relcycle_0)
|
|
868 |
next
|
|
869 |
fix c
|
|
870 |
assume c: "c \<in> singular_relcycle_set (Suc n) X S \<inter>
|
|
871 |
{c. singular_relboundary n (subtopology X S) {} (chain_boundary (Suc n) c)}"
|
|
872 |
then obtain d where d: "singular_chain (Suc n) (subtopology X S) d"
|
|
873 |
"chain_boundary (Suc n) d = chain_boundary (Suc n) c"
|
|
874 |
by (auto simp: singular_boundary)
|
|
875 |
with c have "c - d \<in> singular_relcycle_set (Suc n) X {}"
|
|
876 |
by (auto simp: singular_cycle chain_boundary_diff singular_chain_subtopology singular_relcycle singular_chain_diff)
|
|
877 |
moreover have "homologous_rel_set (Suc n) X S c = homologous_rel_set (Suc n) X S (c - d)"
|
|
878 |
proof (simp add: homologous_rel_set_eq)
|
|
879 |
show "homologous_rel (Suc n) X S c (c - d)"
|
|
880 |
using d by (simp add: homologous_rel_def singular_chain_imp_relboundary)
|
|
881 |
qed
|
|
882 |
ultimately show "\<exists>y. y \<in> singular_relcycle_set (Suc n) X {} \<and>
|
|
883 |
homologous_rel_set (Suc n) X S c = homologous_rel_set (Suc n) X S y"
|
|
884 |
by blast
|
|
885 |
qed
|
|
886 |
also have "\<dots> = ?rhs"
|
|
887 |
by (rule 1) (simp add: hom_boundary_chain_boundary homologous_rel_set_eq_relboundary del: of_nat_Suc)
|
|
888 |
finally show "?lhs = ?rhs" .
|
|
889 |
qed
|
|
890 |
with Suc show ?thesis
|
|
891 |
unfolding carrier_relative_homology_group image_comp id_def by auto
|
|
892 |
qed
|
|
893 |
then show ?thesis
|
|
894 |
by (auto simp: kernel_def int)
|
|
895 |
qed
|
|
896 |
then show ?thesis
|
|
897 |
using hom_boundary_hom hom_induced_hom
|
|
898 |
by (force simp: group_hom_def group_hom_axioms_def)
|
|
899 |
qed
|
|
900 |
|
|
901 |
|
|
902 |
lemma homology_exactness_axiom_2:
|
|
903 |
"exact_seq ([homology_group (p-1) X, homology_group (p-1) (subtopology X S), relative_homology_group p X S],
|
|
904 |
[hom_induced (p-1) (subtopology X S) {} X {} id, hom_boundary p X S])"
|
|
905 |
proof -
|
|
906 |
consider (neg) "p \<le> 0" | (int) n where "p = int (Suc n)"
|
|
907 |
by (metis linear not0_implies_Suc of_nat_0 zero_le_imp_eq_int)
|
|
908 |
then have "kernel (relative_homology_group (p - 1) (subtopology X S) {})
|
|
909 |
(relative_homology_group (p - 1) X {})
|
|
910 |
(hom_induced (p - 1) (subtopology X S) {} X {} id)
|
|
911 |
= hom_boundary p X S ` carrier (relative_homology_group p X S)"
|
|
912 |
proof cases
|
|
913 |
case neg
|
|
914 |
obtain x where "x \<in> carrier (relative_homology_group p X S)"
|
|
915 |
using group_relative_homology_group group.is_monoid by blast
|
|
916 |
with neg show ?thesis
|
|
917 |
unfolding kernel_def singleton_group_def relative_homology_group_def
|
|
918 |
by (force simp: hom_induced_trivial hom_boundary_trivial)
|
|
919 |
next
|
|
920 |
case int
|
|
921 |
have "hom_boundary (int (Suc n)) X S ` carrier (relative_homology_group (int (Suc n)) X S)
|
|
922 |
= carrier (relative_homology_group n (subtopology X S) {}) \<inter>
|
|
923 |
{c. hom_induced n (subtopology X S) {} X {} id c =
|
|
924 |
\<one>\<^bsub>relative_homology_group n X {}\<^esub>}"
|
|
925 |
(is "?lhs = ?rhs")
|
|
926 |
proof -
|
|
927 |
have 1: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<longleftrightarrow> x \<in> C) \<Longrightarrow> f ` (A \<inter> B) = f ` (A \<inter> C)" for f A B C
|
|
928 |
by blast
|
|
929 |
have 2: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<longleftrightarrow> x \<in> f -` C) \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> C" for f A B C
|
|
930 |
by blast
|
|
931 |
have "?lhs = homologous_rel_set n (subtopology X S) {}
|
|
932 |
` (chain_boundary (Suc n) ` singular_relcycle_set (Suc n) X S)"
|
|
933 |
unfolding carrier_relative_homology_group image_comp
|
|
934 |
by (rule image_cong [OF refl]) (simp add: o_def hom_boundary_chain_boundary del: of_nat_Suc)
|
|
935 |
also have "\<dots> = homologous_rel_set n (subtopology X S) {} `
|
|
936 |
(singular_relcycle_set n (subtopology X S) {} \<inter> singular_relboundary_set n X {})"
|
|
937 |
by (force simp: singular_relcycle singular_boundary chain_boundary_boundary_alt)
|
|
938 |
also have "\<dots> = ?rhs"
|
|
939 |
unfolding carrier_relative_homology_group vimage_def
|
|
940 |
apply (rule 2)
|
|
941 |
apply (auto simp: hom_induced_chain_map chain_map_ident homologous_rel_set_eq_relboundary singular_relcycle)
|
|
942 |
done
|
|
943 |
finally show ?thesis .
|
|
944 |
qed
|
|
945 |
then show ?thesis
|
|
946 |
by (auto simp: kernel_def int)
|
|
947 |
qed
|
|
948 |
then show ?thesis
|
|
949 |
using hom_boundary_hom hom_induced_hom
|
|
950 |
by (force simp: group_hom_def group_hom_axioms_def)
|
|
951 |
qed
|
|
952 |
|
|
953 |
|
|
954 |
lemma homology_exactness_axiom_3:
|
|
955 |
"exact_seq ([relative_homology_group p X S, homology_group p X, homology_group p (subtopology X S)],
|
|
956 |
[hom_induced p X {} X S id, hom_induced p (subtopology X S) {} X {} id])"
|
|
957 |
proof (cases "p < 0")
|
|
958 |
case True
|
|
959 |
then show ?thesis
|
|
960 |
apply (simp add: relative_homology_group_def hom_induced_trivial group_hom_def group_hom_axioms_def)
|
|
961 |
apply (auto simp: kernel_def singleton_group_def)
|
|
962 |
done
|
|
963 |
next
|
|
964 |
case False
|
|
965 |
then obtain n where peq: "p = int n"
|
|
966 |
by (metis int_ops(1) linorder_neqE_linordered_idom pos_int_cases)
|
|
967 |
have "hom_induced n (subtopology X S) {} X {} id `
|
|
968 |
(homologous_rel_set n (subtopology X S) {} `
|
|
969 |
singular_relcycle_set n (subtopology X S) {})
|
|
970 |
= {c \<in> homologous_rel_set n X {} ` singular_relcycle_set n X {}.
|
|
971 |
hom_induced n X {} X S id c = singular_relboundary_set n X S}"
|
|
972 |
(is "?lhs = ?rhs")
|
|
973 |
proof -
|
|
974 |
have 2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> f x = f y; \<And>x. x \<in> B \<Longrightarrow> \<exists>y. y \<in> A \<and> f x = f y\<rbrakk>
|
|
975 |
\<Longrightarrow> f ` A = f ` B" for f A B
|
|
976 |
by blast
|
|
977 |
have "?lhs = homologous_rel_set n X {} ` (singular_relcycle_set n (subtopology X S) {})"
|
|
978 |
unfolding image_comp o_def
|
|
979 |
apply (rule image_cong [OF refl])
|
|
980 |
apply (simp add: hom_induced_chain_map singular_relcycle)
|
|
981 |
apply (metis chain_map_ident)
|
|
982 |
done
|
|
983 |
also have "\<dots> = homologous_rel_set n X {} ` (singular_relcycle_set n X {} \<inter> singular_relboundary_set n X S)"
|
|
984 |
proof (rule 2)
|
|
985 |
fix c
|
|
986 |
assume "c \<in> singular_relcycle_set n (subtopology X S) {}"
|
|
987 |
then show "\<exists>y. y \<in> singular_relcycle_set n X {} \<inter> singular_relboundary_set n X S \<and>
|
|
988 |
homologous_rel_set n X {} c = homologous_rel_set n X {} y"
|
|
989 |
using singular_chain_imp_relboundary singular_cycle singular_relboundary_imp_chain singular_relcycle by fastforce
|
|
990 |
next
|
|
991 |
fix c
|
|
992 |
assume "c \<in> singular_relcycle_set n X {} \<inter> singular_relboundary_set n X S"
|
|
993 |
then obtain d e where c: "singular_relcycle n X {} c" "singular_relboundary n X S c"
|
|
994 |
and d: "singular_chain n (subtopology X S) d"
|
|
995 |
and e: "singular_chain (Suc n) X e" "chain_boundary (Suc n) e = c + d"
|
|
996 |
using singular_relboundary_alt by blast
|
|
997 |
then have "chain_boundary n (c + d) = 0"
|
|
998 |
using chain_boundary_boundary_alt by fastforce
|
|
999 |
then have "chain_boundary n c + chain_boundary n d = 0"
|
|
1000 |
by (metis chain_boundary_add)
|
|
1001 |
with c have "singular_relcycle n (subtopology X S) {} (- d)"
|
|
1002 |
by (metis (no_types) d eq_add_iff singular_cycle singular_relcycle_minus)
|
|
1003 |
moreover have "homologous_rel n X {} c (- d)"
|
|
1004 |
using c
|
|
1005 |
by (metis diff_minus_eq_add e homologous_rel_def singular_boundary)
|
|
1006 |
ultimately
|
|
1007 |
show "\<exists>y. y \<in> singular_relcycle_set n (subtopology X S) {} \<and>
|
|
1008 |
homologous_rel_set n X {} c = homologous_rel_set n X {} y"
|
|
1009 |
by (force simp: homologous_rel_set_eq)
|
|
1010 |
qed
|
|
1011 |
also have "\<dots> = homologous_rel_set n X {} `
|
|
1012 |
(singular_relcycle_set n X {} \<inter> homologous_rel_set n X {} -` {x. hom_induced n X {} X S id x = singular_relboundary_set n X S})"
|
|
1013 |
by (rule 2) (auto simp: hom_induced_chain_map homologous_rel_set_eq_relboundary chain_map_ident [of _ X] singular_cycle cong: conj_cong)
|
|
1014 |
also have "\<dots> = ?rhs"
|
|
1015 |
by blast
|
|
1016 |
finally show ?thesis .
|
|
1017 |
qed
|
|
1018 |
then have "kernel (relative_homology_group p X {}) (relative_homology_group p X S) (hom_induced p X {} X S id)
|
|
1019 |
= hom_induced p (subtopology X S) {} X {} id ` carrier (relative_homology_group p (subtopology X S) {})"
|
|
1020 |
by (simp add: kernel_def carrier_relative_homology_group peq)
|
|
1021 |
then show ?thesis
|
|
1022 |
by (simp add: not_less group_hom_def group_hom_axioms_def hom_induced_hom)
|
|
1023 |
qed
|
|
1024 |
|
|
1025 |
|
|
1026 |
lemma homology_dimension_axiom:
|
|
1027 |
assumes X: "topspace X = {a}" and "p \<noteq> 0"
|
|
1028 |
shows "trivial_group(homology_group p X)"
|
|
1029 |
proof (cases "p < 0")
|
|
1030 |
case True
|
|
1031 |
then show ?thesis
|
|
1032 |
by simp
|
|
1033 |
next
|
|
1034 |
case False
|
|
1035 |
then obtain n where peq: "p = int n" "n > 0"
|
|
1036 |
by (metis assms(2) neq0_conv nonneg_int_cases not_less of_nat_0)
|
|
1037 |
have "homologous_rel_set n X {} ` singular_relcycle_set n X {} = {singular_relcycle_set n X {}}"
|
|
1038 |
(is "?lhs = ?rhs")
|
|
1039 |
proof
|
|
1040 |
show "?lhs \<subseteq> ?rhs"
|
|
1041 |
using peq assms
|
|
1042 |
by (auto simp: image_subset_iff homologous_rel_set_eq_relboundary simp flip: singular_boundary_set_eq_cycle_singleton)
|
|
1043 |
have "singular_relboundary n X {} 0"
|
|
1044 |
by simp
|
|
1045 |
with peq assms
|
|
1046 |
show "?rhs \<subseteq> ?lhs"
|
|
1047 |
by (auto simp: image_iff simp flip: homologous_rel_eq_relboundary singular_boundary_set_eq_cycle_singleton)
|
|
1048 |
qed
|
|
1049 |
with peq assms show ?thesis
|
|
1050 |
unfolding trivial_group_def
|
|
1051 |
by (simp add: carrier_relative_homology_group singular_boundary_set_eq_cycle_singleton [OF X])
|
|
1052 |
qed
|
|
1053 |
|
|
1054 |
|
|
1055 |
proposition homology_homotopy_axiom:
|
|
1056 |
assumes "homotopic_with (\<lambda>h. h ` S \<subseteq> T) X Y f g"
|
|
1057 |
shows "hom_induced p X S Y T f = hom_induced p X S Y T g"
|
|
1058 |
proof (cases "p < 0")
|
|
1059 |
case True
|
|
1060 |
then show ?thesis
|
|
1061 |
by (simp add: hom_induced_trivial)
|
|
1062 |
next
|
|
1063 |
case False
|
|
1064 |
then obtain n where peq: "p = int n"
|
|
1065 |
by (metis int_nat_eq not_le)
|
|
1066 |
have cont: "continuous_map X Y f" "continuous_map X Y g"
|
|
1067 |
using assms homotopic_with_imp_continuous_maps by blast+
|
|
1068 |
have im: "f ` (topspace X \<inter> S) \<subseteq> T" "g ` (topspace X \<inter> S) \<subseteq> T"
|
|
1069 |
using homotopic_with_imp_property assms by blast+
|
|
1070 |
show ?thesis
|
|
1071 |
proof
|
|
1072 |
fix c show "hom_induced p X S Y T f c = hom_induced p X S Y T g c"
|
|
1073 |
proof (cases "c \<in> carrier(relative_homology_group p X S)")
|
|
1074 |
case True
|
|
1075 |
then obtain a where a: "c = homologous_rel_set n X S a" "singular_relcycle n X S a"
|
|
1076 |
unfolding carrier_relative_homology_group peq by auto
|
|
1077 |
then show ?thesis
|
|
1078 |
apply (simp add: peq hom_induced_chain_map_gen cont im homologous_rel_set_eq)
|
|
1079 |
apply (blast intro: assms homotopic_imp_homologous_rel_chain_maps)
|
|
1080 |
done
|
|
1081 |
qed (simp add: hom_induced_default)
|
|
1082 |
qed
|
|
1083 |
qed
|
|
1084 |
|
|
1085 |
proposition homology_excision_axiom:
|
|
1086 |
assumes "X closure_of U \<subseteq> X interior_of T" "T \<subseteq> S"
|
|
1087 |
shows
|
|
1088 |
"hom_induced p (subtopology X (S - U)) (T - U) (subtopology X S) T id
|
|
1089 |
\<in> iso (relative_homology_group p (subtopology X (S - U)) (T - U))
|
|
1090 |
(relative_homology_group p (subtopology X S) T)"
|
|
1091 |
proof (cases "p < 0")
|
|
1092 |
case True
|
|
1093 |
then show ?thesis
|
|
1094 |
unfolding iso_def bij_betw_def relative_homology_group_def by (simp add: hom_induced_trivial)
|
|
1095 |
next
|
|
1096 |
case False
|
|
1097 |
then obtain n where peq: "p = int n"
|
|
1098 |
by (metis int_nat_eq not_le)
|
|
1099 |
have cont: "continuous_map (subtopology X (S - U)) (subtopology X S) id"
|
|
1100 |
by (simp add: closure_of_subtopology_mono continuous_map_eq_image_closure_subset)
|
|
1101 |
have TU: "topspace X \<inter> (S - U) \<inter> (T - U) \<subseteq> T"
|
|
1102 |
by auto
|
|
1103 |
show ?thesis
|
|
1104 |
proof (simp add: iso_def peq carrier_relative_homology_group bij_betw_def hom_induced_hom, intro conjI)
|
|
1105 |
show "inj_on (hom_induced n (subtopology X (S - U)) (T - U) (subtopology X S) T id)
|
|
1106 |
(homologous_rel_set n (subtopology X (S - U)) (T - U) `
|
|
1107 |
singular_relcycle_set n (subtopology X (S - U)) (T - U))"
|
|
1108 |
unfolding inj_on_def
|
|
1109 |
proof (clarsimp simp add: homologous_rel_set_eq)
|
|
1110 |
fix c d
|
|
1111 |
assume c: "singular_relcycle n (subtopology X (S - U)) (T - U) c"
|
|
1112 |
and d: "singular_relcycle n (subtopology X (S - U)) (T - U) d"
|
|
1113 |
and hh: "hom_induced n (subtopology X (S - U)) (T - U) (subtopology X S) T id
|
|
1114 |
(homologous_rel_set n (subtopology X (S - U)) (T - U) c)
|
|
1115 |
= hom_induced n (subtopology X (S - U)) (T - U) (subtopology X S) T id
|
|
1116 |
(homologous_rel_set n (subtopology X (S - U)) (T - U) d)"
|
|
1117 |
then have scc: "singular_chain n (subtopology X (S - U)) c"
|
|
1118 |
and scd: "singular_chain n (subtopology X (S - U)) d"
|
|
1119 |
using singular_relcycle by blast+
|
|
1120 |
have "singular_relboundary n (subtopology X (S - U)) (T - U) c"
|
|
1121 |
if srb: "singular_relboundary n (subtopology X S) T c"
|
|
1122 |
and src: "singular_relcycle n (subtopology X (S - U)) (T - U) c" for c
|
|
1123 |
proof -
|
|
1124 |
have [simp]: "(S - U) \<inter> (T - U) = T - U" "S \<inter> T = T"
|
|
1125 |
using \<open>T \<subseteq> S\<close> by blast+
|
|
1126 |
have c: "singular_chain n (subtopology X (S - U)) c"
|
|
1127 |
"singular_chain (n - Suc 0) (subtopology X (T - U)) (chain_boundary n c)"
|
|
1128 |
using that by (auto simp: singular_relcycle_def mod_subset_def subtopology_subtopology)
|
|
1129 |
obtain d e where d: "singular_chain (Suc n) (subtopology X S) d"
|
|
1130 |
and e: "singular_chain n (subtopology X T) e"
|
|
1131 |
and dce: "chain_boundary (Suc n) d = c + e"
|
|
1132 |
using srb by (auto simp: singular_relboundary_alt subtopology_subtopology)
|
|
1133 |
obtain m f g where f: "singular_chain (Suc n) (subtopology X (S - U)) f"
|
|
1134 |
and g: "singular_chain (Suc n) (subtopology X T) g"
|
|
1135 |
and dfg: "(singular_subdivision (Suc n) ^^ m) d = f + g"
|
|
1136 |
using excised_chain_exists [OF assms d] .
|
|
1137 |
obtain h where
|
|
1138 |
h0: "\<And>p. h p 0 = (0 :: 'a chain)"
|
|
1139 |
and hdiff: "\<And>p c1 c2. h p (c1-c2) = h p c1 - h p c2"
|
|
1140 |
and hSuc: "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"
|
|
1141 |
and hchain: "\<And>p X c. singular_chain p X c
|
|
1142 |
\<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c)
|
|
1143 |
= (singular_subdivision p ^^ m) c - c"
|
|
1144 |
using chain_homotopic_iterated_singular_subdivision by blast
|
|
1145 |
have hadd: "\<And>p c1 c2. h p (c1 + c2) = h p c1 + h p c2"
|
|
1146 |
by (metis add_diff_cancel diff_add_cancel hdiff)
|
|
1147 |
define c1 where "c1 \<equiv> f - h n c"
|
|
1148 |
define c2 where "c2 \<equiv> chain_boundary (Suc n) (h n e) - (chain_boundary (Suc n) g - e)"
|
|
1149 |
show ?thesis
|
|
1150 |
unfolding singular_relboundary_alt
|
|
1151 |
proof (intro exI conjI)
|
|
1152 |
show c1: "singular_chain (Suc n) (subtopology X (S - U)) c1"
|
|
1153 |
by (simp add: \<open>singular_chain n (subtopology X (S - U)) c\<close> c1_def f hSuc singular_chain_diff)
|
|
1154 |
have "chain_boundary (Suc n) (chain_boundary (Suc (Suc n)) (h (Suc n) d) + h n (c+e))
|
|
1155 |
= chain_boundary (Suc n) (f + g - d)"
|
|
1156 |
using hchain [OF d] by (simp add: dce dfg)
|
|
1157 |
then have "chain_boundary (Suc n) (h n (c + e))
|
|
1158 |
= chain_boundary (Suc n) f + chain_boundary (Suc n) g - (c + e)"
|
|
1159 |
using chain_boundary_boundary_alt [of "Suc n" "subtopology X S"]
|
|
1160 |
by (simp add: chain_boundary_add chain_boundary_diff d hSuc dce)
|
|
1161 |
then have "chain_boundary (Suc n) (h n c) + chain_boundary (Suc n) (h n e)
|
|
1162 |
= chain_boundary (Suc n) f + chain_boundary (Suc n) g - (c + e)"
|
|
1163 |
by (simp add: chain_boundary_add hadd)
|
|
1164 |
then have *: "chain_boundary (Suc n) (f - h n c) = c + (chain_boundary (Suc n) (h n e) - (chain_boundary (Suc n) g - e))"
|
|
1165 |
by (simp add: algebra_simps chain_boundary_diff)
|
|
1166 |
then show "chain_boundary (Suc n) c1 = c + c2"
|
|
1167 |
unfolding c1_def c2_def
|
|
1168 |
by (simp add: algebra_simps chain_boundary_diff)
|
|
1169 |
have "singular_chain n (subtopology X (S - U)) c2" "singular_chain n (subtopology X T) c2"
|
|
1170 |
using singular_chain_diff c c1 *
|
|
1171 |
unfolding c1_def c2_def
|
|
1172 |
apply (metis add_diff_cancel_left' singular_chain_boundary_alt)
|
|
1173 |
by (simp add: e g hSuc singular_chain_boundary_alt singular_chain_diff)
|
|
1174 |
then show "singular_chain n (subtopology (subtopology X (S - U)) (T - U)) c2"
|
|
1175 |
by (fastforce simp add: singular_chain_subtopology)
|
|
1176 |
qed
|
|
1177 |
qed
|
|
1178 |
then have "singular_relboundary n (subtopology X S) T (c - d) \<Longrightarrow>
|
|
1179 |
singular_relboundary n (subtopology X (S - U)) (T - U) (c - d)"
|
|
1180 |
using c d singular_relcycle_diff by metis
|
|
1181 |
with hh show "homologous_rel n (subtopology X (S - U)) (T - U) c d"
|
|
1182 |
apply (simp add: hom_induced_chain_map cont c d chain_map_ident [OF scc] chain_map_ident [OF scd])
|
|
1183 |
using homologous_rel_set_eq homologous_rel_def by metis
|
|
1184 |
qed
|
|
1185 |
next
|
|
1186 |
have h: "homologous_rel_set n (subtopology X S) T a
|
|
1187 |
\<in> (\<lambda>x. homologous_rel_set n (subtopology X S) T (chain_map n id x)) `
|
|
1188 |
singular_relcycle_set n (subtopology X (S - U)) (T - U)"
|
|
1189 |
if a: "singular_relcycle n (subtopology X S) T a" for a
|
|
1190 |
proof -
|
|
1191 |
obtain c' where c': "singular_relcycle n (subtopology X (S - U)) (T - U) c'"
|
|
1192 |
"homologous_rel n (subtopology X S) T a c'"
|
|
1193 |
using a by (blast intro: excised_relcycle_exists [OF assms])
|
|
1194 |
then have scc': "singular_chain n (subtopology X S) c'"
|
|
1195 |
using homologous_rel_singular_chain singular_relcycle that by blast
|
|
1196 |
then show ?thesis
|
|
1197 |
apply (rule_tac x="c'" in image_eqI)
|
|
1198 |
apply (auto simp: scc' chain_map_ident [of _ "subtopology X S"] c' homologous_rel_set_eq)
|
|
1199 |
done
|
|
1200 |
qed
|
|
1201 |
show "hom_induced n (subtopology X (S - U)) (T - U) (subtopology X S) T id `
|
|
1202 |
homologous_rel_set n (subtopology X (S - U)) (T - U) `
|
|
1203 |
singular_relcycle_set n (subtopology X (S - U)) (T - U)
|
|
1204 |
= homologous_rel_set n (subtopology X S) T ` singular_relcycle_set n (subtopology X S) T"
|
|
1205 |
apply (simp add: image_comp o_def hom_induced_chain_map_gen cont TU topspace_subtopology
|
|
1206 |
cong: image_cong_simp)
|
|
1207 |
apply (force simp: cont h singular_relcycle_chain_map)
|
|
1208 |
done
|
|
1209 |
qed
|
|
1210 |
qed
|
|
1211 |
|
|
1212 |
|
|
1213 |
subsection\<open>Additivity axiom\<close>
|
|
1214 |
|
|
1215 |
text\<open>Not in the original Eilenberg-Steenrod list but usually included nowadays,
|
|
1216 |
following Milnor's "On Axiomatic Homology Theory".\<close>
|
|
1217 |
|
|
1218 |
lemma iso_chain_group_sum:
|
|
1219 |
assumes disj: "pairwise disjnt \<U>" and UU: "\<Union>\<U> = topspace X"
|
|
1220 |
and subs: "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; ~ disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T"
|
|
1221 |
shows "(\<lambda>f. sum' f \<U>) \<in> iso (sum_group \<U> (\<lambda>S. chain_group p (subtopology X S))) (chain_group p X)"
|
|
1222 |
proof -
|
|
1223 |
have pw: "pairwise (\<lambda>i j. disjnt (singular_simplex_set p (subtopology X i))
|
|
1224 |
(singular_simplex_set p (subtopology X j))) \<U>"
|
|
1225 |
proof
|
|
1226 |
fix S T
|
|
1227 |
assume "S \<in> \<U>" "T \<in> \<U>" "S \<noteq> T"
|
|
1228 |
then show "disjnt (singular_simplex_set p (subtopology X S))
|
|
1229 |
(singular_simplex_set p (subtopology X T))"
|
|
1230 |
using nonempty_standard_simplex [of p] disj
|
|
1231 |
by (fastforce simp: pairwise_def disjnt_def singular_simplex_subtopology image_subset_iff)
|
|
1232 |
qed
|
|
1233 |
have "\<exists>S\<in>\<U>. singular_simplex p (subtopology X S) f"
|
|
1234 |
if f: "singular_simplex p X f" for f
|
|
1235 |
proof -
|
|
1236 |
obtain x where x: "x \<in> topspace X" "x \<in> f ` standard_simplex p"
|
|
1237 |
using f nonempty_standard_simplex [of p] continuous_map_image_subset_topspace
|
|
1238 |
unfolding singular_simplex_def by fastforce
|
|
1239 |
then obtain S where "S \<in> \<U>" "x \<in> S"
|
|
1240 |
using UU by auto
|
|
1241 |
have "f ` standard_simplex p \<subseteq> S"
|
|
1242 |
proof (rule subs)
|
|
1243 |
have cont: "continuous_map (subtopology (powertop_real UNIV)
|
|
1244 |
(standard_simplex p)) X f"
|
|
1245 |
using f singular_simplex_def by auto
|
|
1246 |
show "compactin X (f ` standard_simplex p)"
|
|
1247 |
by (simp add: compactin_subtopology compactin_standard_simplex image_compactin [OF _ cont])
|
|
1248 |
show "path_connectedin X (f ` standard_simplex p)"
|
|
1249 |
by (simp add: path_connectedin_subtopology path_connectedin_standard_simplex path_connectedin_continuous_map_image [OF cont])
|
|
1250 |
have "standard_simplex p \<noteq> {}"
|
|
1251 |
by (simp add: nonempty_standard_simplex)
|
|
1252 |
then
|
|
1253 |
show "\<not> disjnt (f ` standard_simplex p) S"
|
|
1254 |
using x \<open>x \<in> S\<close> by (auto simp: disjnt_def)
|
|
1255 |
qed (auto simp: \<open>S \<in> \<U>\<close>)
|
|
1256 |
then show ?thesis
|
|
1257 |
by (meson \<open>S \<in> \<U>\<close> singular_simplex_subtopology that)
|
|
1258 |
qed
|
|
1259 |
then have "(\<Union>i\<in>\<U>. singular_simplex_set p (subtopology X i)) = singular_simplex_set p X"
|
|
1260 |
by (auto simp: singular_simplex_subtopology)
|
|
1261 |
then show ?thesis
|
|
1262 |
using iso_free_Abelian_group_sum [OF pw] by (simp add: chain_group_def)
|
|
1263 |
qed
|
|
1264 |
|
|
1265 |
lemma relcycle_group_0_eq_chain_group: "relcycle_group 0 X {} = chain_group 0 X"
|
|
1266 |
apply (rule monoid.equality, simp)
|
|
1267 |
apply (simp_all add: relcycle_group_def chain_group_def)
|
|
1268 |
by (metis chain_boundary_def singular_cycle)
|
|
1269 |
|
|
1270 |
|
|
1271 |
proposition iso_cycle_group_sum:
|
|
1272 |
assumes disj: "pairwise disjnt \<U>" and UU: "\<Union>\<U> = topspace X"
|
|
1273 |
and subs: "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; \<not> disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T"
|
|
1274 |
shows "(\<lambda>f. sum' f \<U>) \<in> iso (sum_group \<U> (\<lambda>T. relcycle_group p (subtopology X T) {}))
|
|
1275 |
(relcycle_group p X {})"
|
|
1276 |
proof (cases "p = 0")
|
|
1277 |
case True
|
|
1278 |
then show ?thesis
|
|
1279 |
by (simp add: relcycle_group_0_eq_chain_group iso_chain_group_sum [OF assms])
|
|
1280 |
next
|
|
1281 |
case False
|
|
1282 |
let ?SG = "(sum_group \<U> (\<lambda>T. chain_group p (subtopology X T)))"
|
|
1283 |
let ?PI = "(\<Pi>\<^sub>E T\<in>\<U>. singular_relcycle_set p (subtopology X T) {})"
|
|
1284 |
have "(\<lambda>f. sum' f \<U>) \<in> Group.iso (subgroup_generated ?SG (carrier ?SG \<inter> ?PI))
|
|
1285 |
(subgroup_generated (chain_group p X) (singular_relcycle_set p X {}))"
|
|
1286 |
proof (rule group_hom.iso_between_subgroups)
|
|
1287 |
have iso: "(\<lambda>f. sum' f \<U>) \<in> Group.iso ?SG (chain_group p X)"
|
|
1288 |
by (auto simp: assms iso_chain_group_sum)
|
|
1289 |
then show "group_hom ?SG (chain_group p X) (\<lambda>f. sum' f \<U>)"
|
|
1290 |
by (auto simp: iso_imp_homomorphism group_hom_def group_hom_axioms_def)
|
|
1291 |
have B: "sum' f \<U> \<in> singular_relcycle_set p X {} \<longleftrightarrow> f \<in> (carrier ?SG \<inter> ?PI)"
|
|
1292 |
if "f \<in> (carrier ?SG)" for f
|
|
1293 |
proof -
|
|
1294 |
have f: "\<And>S. S \<in> \<U> \<longrightarrow> singular_chain p (subtopology X S) (f S)"
|
|
1295 |
"f \<in> extensional \<U>" "finite {i \<in> \<U>. f i \<noteq> 0}"
|
|
1296 |
using that by (auto simp: carrier_sum_group PiE_def Pi_def)
|
|
1297 |
then have rfin: "finite {S \<in> \<U>. restrict (chain_boundary p \<circ> f) \<U> S \<noteq> 0}"
|
|
1298 |
by (auto elim: rev_finite_subset)
|
|
1299 |
have "chain_boundary p ((\<Sum>x | x \<in> \<U> \<and> f x \<noteq> 0. f x)) = 0
|
|
1300 |
\<longleftrightarrow> (\<forall>S \<in> \<U>. chain_boundary p (f S) = 0)" (is "?cb = 0 \<longleftrightarrow> ?rhs")
|
|
1301 |
proof
|
|
1302 |
assume "?cb = 0"
|
|
1303 |
moreover have "?cb = sum' (\<lambda>S. chain_boundary p (f S)) \<U>"
|
|
1304 |
unfolding sum.G_def using rfin f
|
|
1305 |
by (force simp: chain_boundary_sum intro: sum.mono_neutral_right cong: conj_cong)
|
|
1306 |
ultimately have eq0: "sum' (\<lambda>S. chain_boundary p (f S)) \<U> = 0"
|
|
1307 |
by simp
|
|
1308 |
have "(\<lambda>f. sum' f \<U>) \<in> hom (sum_group \<U> (\<lambda>S. chain_group (p - Suc 0) (subtopology X S)))
|
|
1309 |
(chain_group (p - Suc 0) X)"
|
|
1310 |
and inj: "inj_on (\<lambda>f. sum' f \<U>) (carrier (sum_group \<U> (\<lambda>S. chain_group (p - Suc 0) (subtopology X S))))"
|
|
1311 |
using iso_chain_group_sum [OF assms, of "p-1"] by (auto simp: iso_def bij_betw_def)
|
|
1312 |
then have eq: "\<lbrakk>f \<in> (\<Pi>\<^sub>E i\<in>\<U>. singular_chain_set (p - Suc 0) (subtopology X i));
|
|
1313 |
finite {S \<in> \<U>. f S \<noteq> 0}; sum' f \<U> = 0; S \<in> \<U>\<rbrakk> \<Longrightarrow> f S = 0" for f S
|
|
1314 |
apply (simp add: group_hom_def group_hom_axioms_def group_hom.inj_on_one_iff [of _ "chain_group (p-1) X"])
|
|
1315 |
apply (auto simp: carrier_sum_group fun_eq_iff that)
|
|
1316 |
done
|
|
1317 |
show ?rhs
|
|
1318 |
proof clarify
|
|
1319 |
fix S assume "S \<in> \<U>"
|
|
1320 |
then show "chain_boundary p (f S) = 0"
|
|
1321 |
using eq [of "restrict (chain_boundary p \<circ> f) \<U>" S] rfin f eq0
|
|
1322 |
by (simp add: singular_chain_boundary cong: conj_cong)
|
|
1323 |
qed
|
|
1324 |
next
|
|
1325 |
assume ?rhs
|
|
1326 |
then show "?cb = 0"
|
|
1327 |
by (force simp: chain_boundary_sum intro: sum.mono_neutral_right)
|
|
1328 |
qed
|
|
1329 |
moreover
|
|
1330 |
have "(\<And>S. S \<in> \<U> \<longrightarrow> singular_chain p (subtopology X S) (f S))
|
|
1331 |
\<Longrightarrow> singular_chain p X (\<Sum>x | x \<in> \<U> \<and> f x \<noteq> 0. f x)"
|
|
1332 |
by (metis (no_types, lifting) mem_Collect_eq singular_chain_subtopology singular_chain_sum)
|
|
1333 |
ultimately show ?thesis
|
|
1334 |
using f by (auto simp: carrier_sum_group sum.G_def singular_cycle PiE_iff)
|
|
1335 |
qed
|
|
1336 |
have "singular_relcycle_set p X {} \<subseteq> carrier (chain_group p X)"
|
|
1337 |
using subgroup.subset subgroup_singular_relcycle by blast
|
|
1338 |
then show "(\<lambda>f. sum' f \<U>) ` (carrier ?SG \<inter> ?PI) = singular_relcycle_set p X {}"
|
|
1339 |
using iso B
|
|
1340 |
apply (auto simp: iso_def bij_betw_def)
|
|
1341 |
apply (force simp: singular_relcycle)
|
|
1342 |
done
|
|
1343 |
qed (auto simp: assms iso_chain_group_sum)
|
|
1344 |
then show ?thesis
|
|
1345 |
by (simp add: relcycle_group_def sum_group_subgroup_generated subgroup_singular_relcycle)
|
|
1346 |
qed
|
|
1347 |
|
|
1348 |
|
|
1349 |
proposition homology_additivity_axiom_gen:
|
|
1350 |
assumes disj: "pairwise disjnt \<U>" and UU: "\<Union>\<U> = topspace X"
|
|
1351 |
and subs: "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; \<not> disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T"
|
|
1352 |
shows "(\<lambda>x. gfinprod (homology_group p X)
|
|
1353 |
(\<lambda>V. hom_induced p (subtopology X V) {} X {} id (x V)) \<U>)
|
|
1354 |
\<in> iso (sum_group \<U> (\<lambda>S. homology_group p (subtopology X S))) (homology_group p X)"
|
|
1355 |
(is "?h \<in> iso ?SG ?HG")
|
|
1356 |
proof (cases "p < 0")
|
|
1357 |
case True
|
|
1358 |
then have [simp]: "gfinprod (singleton_group undefined) (\<lambda>v. undefined) \<U> = undefined"
|
|
1359 |
by (metis Pi_I carrier_singleton_group comm_group_def comm_monoid.gfinprod_closed singletonD singleton_abelian_group)
|
|
1360 |
show ?thesis
|
|
1361 |
using True
|
|
1362 |
apply (simp add: iso_def relative_homology_group_def hom_induced_trivial carrier_sum_group)
|
|
1363 |
apply (auto simp: singleton_group_def bij_betw_def inj_on_def fun_eq_iff)
|
|
1364 |
done
|
|
1365 |
next
|
|
1366 |
case False
|
|
1367 |
then obtain n where peq: "p = int n"
|
|
1368 |
by (metis int_ops(1) linorder_neqE_linordered_idom pos_int_cases)
|
|
1369 |
interpret comm_group "homology_group p X"
|
|
1370 |
by (rule abelian_homology_group)
|
|
1371 |
show ?thesis
|
|
1372 |
proof (simp add: iso_def bij_betw_def, intro conjI)
|
|
1373 |
show "?h \<in> hom ?SG ?HG"
|
|
1374 |
by (rule hom_group_sum) (simp_all add: hom_induced_hom)
|
|
1375 |
then interpret group_hom ?SG ?HG ?h
|
|
1376 |
by (simp add: group_hom_def group_hom_axioms_def)
|
|
1377 |
have carrSG: "carrier ?SG
|
|
1378 |
= (\<lambda>x. \<lambda>S\<in>\<U>. homologous_rel_set n (subtopology X S) {} (x S))
|
|
1379 |
` (carrier (sum_group \<U> (\<lambda>S. relcycle_group n (subtopology X S) {})))" (is "?lhs = ?rhs")
|
|
1380 |
proof
|
|
1381 |
show "?lhs \<subseteq> ?rhs"
|
|
1382 |
proof (clarsimp simp: carrier_sum_group carrier_relative_homology_group peq)
|
|
1383 |
fix z
|
|
1384 |
assume z: "z \<in> (\<Pi>\<^sub>E S\<in>\<U>. homologous_rel_set n (subtopology X S) {} ` singular_relcycle_set n (subtopology X S) {})"
|
|
1385 |
and fin: "finite {S \<in> \<U>. z S \<noteq> singular_relboundary_set n (subtopology X S) {}}"
|
|
1386 |
then obtain c where c: "\<forall>S\<in>\<U>. singular_relcycle n (subtopology X S) {} (c S)
|
|
1387 |
\<and> z S = homologous_rel_set n (subtopology X S) {} (c S)"
|
|
1388 |
by (simp add: PiE_def Pi_def image_def) metis
|
|
1389 |
let ?f = "\<lambda>S\<in>\<U>. if singular_relboundary n (subtopology X S) {} (c S) then 0 else c S"
|
|
1390 |
have "z = (\<lambda>S\<in>\<U>. homologous_rel_set n (subtopology X S) {} (?f S))"
|
|
1391 |
apply (simp_all add: c fun_eq_iff PiE_arb [OF z])
|
|
1392 |
apply (metis homologous_rel_eq_relboundary singular_boundary singular_relboundary_0)
|
|
1393 |
done
|
|
1394 |
moreover have "?f \<in> (\<Pi>\<^sub>E i\<in>\<U>. singular_relcycle_set n (subtopology X i) {})"
|
|
1395 |
by (simp add: c fun_eq_iff PiE_arb [OF z])
|
|
1396 |
moreover have "finite {i \<in> \<U>. ?f i \<noteq> 0}"
|
|
1397 |
apply (rule finite_subset [OF _ fin])
|
|
1398 |
using z apply (clarsimp simp: PiE_def Pi_def image_def)
|
|
1399 |
by (metis c homologous_rel_set_eq_relboundary singular_boundary)
|
|
1400 |
ultimately
|
|
1401 |
show "z \<in> (\<lambda>x. \<lambda>S\<in>\<U>. homologous_rel_set n (subtopology X S) {} (x S)) `
|
|
1402 |
{x \<in> \<Pi>\<^sub>E i\<in>\<U>. singular_relcycle_set n (subtopology X i) {}. finite {i \<in> \<U>. x i \<noteq> 0}}"
|
|
1403 |
by blast
|
|
1404 |
qed
|
|
1405 |
show "?rhs \<subseteq> ?lhs"
|
|
1406 |
by (force simp: peq carrier_sum_group carrier_relative_homology_group homologous_rel_set_eq_relboundary
|
|
1407 |
elim: rev_finite_subset)
|
|
1408 |
qed
|
|
1409 |
have gf: "gfinprod (homology_group p X)
|
|
1410 |
(\<lambda>V. hom_induced n (subtopology X V) {} X {} id
|
|
1411 |
((\<lambda>S\<in>\<U>. homologous_rel_set n (subtopology X S) {} (z S)) V)) \<U>
|
|
1412 |
= homologous_rel_set n X {} (sum' z \<U>)" (is "?lhs = ?rhs")
|
|
1413 |
if z: "z \<in> carrier (sum_group \<U> (\<lambda>S. relcycle_group n (subtopology X S) {}))" for z
|
|
1414 |
proof -
|
|
1415 |
have hom_pi: "(\<lambda>S. homologous_rel_set n X {} (z S)) \<in> \<U> \<rightarrow> carrier (homology_group p X)"
|
|
1416 |
apply (rule Pi_I)
|
|
1417 |
using z
|
|
1418 |
apply (force simp: peq carrier_sum_group carrier_relative_homology_group singular_chain_subtopology singular_cycle)
|
|
1419 |
done
|
|
1420 |
have fin: "finite {S \<in> \<U>. z S \<noteq> 0}"
|
|
1421 |
using that by (force simp: carrier_sum_group)
|
|
1422 |
have "?lhs = gfinprod (homology_group p X) (\<lambda>S. homologous_rel_set n X {} (z S)) \<U>"
|
|
1423 |
apply (rule gfinprod_cong [OF refl Pi_I])
|
|
1424 |
apply (simp add: hom_induced_carrier peq)
|
|
1425 |
using that
|
|
1426 |
apply (auto simp: peq simp_implies_def carrier_sum_group PiE_def Pi_def chain_map_ident singular_cycle hom_induced_chain_map)
|
|
1427 |
done
|
|
1428 |
also have "\<dots> = gfinprod (homology_group p X)
|
|
1429 |
(\<lambda>S. homologous_rel_set n X {} (z S)) {S \<in> \<U>. z S \<noteq> 0}"
|
|
1430 |
apply (rule gfinprod_mono_neutral_cong_right, simp_all add: hom_pi)
|
|
1431 |
apply (simp add: relative_homology_group_def peq)
|
|
1432 |
apply (metis homologous_rel_eq_relboundary singular_relboundary_0)
|
|
1433 |
done
|
|
1434 |
also have "\<dots> = ?rhs"
|
|
1435 |
proof -
|
|
1436 |
have "gfinprod (homology_group p X) (\<lambda>S. homologous_rel_set n X {} (z S)) \<F>
|
|
1437 |
= homologous_rel_set n X {} (sum z \<F>)"
|
|
1438 |
if "finite \<F>" "\<F> \<subseteq> {S \<in> \<U>. z S \<noteq> 0}" for \<F>
|
|
1439 |
using that
|
|
1440 |
proof (induction \<F>)
|
|
1441 |
case empty
|
|
1442 |
have "\<one>\<^bsub>homology_group p X\<^esub> = homologous_rel_set n X {} 0"
|
|
1443 |
apply (simp add: relative_homology_group_def peq)
|
|
1444 |
by (metis diff_zero homologous_rel_def homologous_rel_sym)
|
|
1445 |
then show ?case
|
|
1446 |
by simp
|
|
1447 |
next
|
|
1448 |
case (insert S \<F>)
|
|
1449 |
with z have pi: "(\<lambda>S. homologous_rel_set n X {} (z S)) \<in> \<F> \<rightarrow> carrier (homology_group p X)"
|
|
1450 |
"homologous_rel_set n X {} (z S) \<in> carrier (homology_group p X)"
|
|
1451 |
by (force simp: peq carrier_sum_group carrier_relative_homology_group singular_chain_subtopology singular_cycle)+
|
|
1452 |
have hom: "homologous_rel_set n X {} (z S) \<in> carrier (homology_group p X)"
|
|
1453 |
using insert z
|
|
1454 |
by (force simp: peq carrier_sum_group carrier_relative_homology_group singular_chain_subtopology singular_cycle)
|
|
1455 |
show ?case
|
|
1456 |
using insert z
|
|
1457 |
proof (simp add: pi)
|
|
1458 |
show "homologous_rel_set n X {} (z S) \<otimes>\<^bsub>homology_group p X\<^esub> homologous_rel_set n X {} (sum z \<F>)
|
|
1459 |
= homologous_rel_set n X {} (z S + sum z \<F>)"
|
|
1460 |
using insert z apply (auto simp: peq homologous_rel_add mult_relative_homology_group)
|
|
1461 |
by (metis (no_types, lifting) diff_add_cancel diff_diff_eq2 homologous_rel_def homologous_rel_refl)
|
|
1462 |
qed
|
|
1463 |
qed
|
|
1464 |
with fin show ?thesis
|
|
1465 |
by (simp add: sum.G_def)
|
|
1466 |
qed
|
|
1467 |
finally show ?thesis .
|
|
1468 |
qed
|
|
1469 |
show "inj_on ?h (carrier ?SG)"
|
|
1470 |
proof (clarsimp simp add: inj_on_one_iff)
|
|
1471 |
fix x
|
|
1472 |
assume x: "x \<in> carrier (sum_group \<U> (\<lambda>S. homology_group p (subtopology X S)))"
|
|
1473 |
and 1: "gfinprod (homology_group p X) (\<lambda>V. hom_induced p (subtopology X V) {} X {} id (x V)) \<U>
|
|
1474 |
= \<one>\<^bsub>homology_group p X\<^esub>"
|
|
1475 |
have feq: "(\<lambda>S\<in>\<U>. homologous_rel_set n (subtopology X S) {} (z S))
|
|
1476 |
= (\<lambda>S\<in>\<U>. \<one>\<^bsub>homology_group p (subtopology X S)\<^esub>)"
|
|
1477 |
if z: "z \<in> carrier (sum_group \<U> (\<lambda>S. relcycle_group n (subtopology X S) {}))"
|
|
1478 |
and eq: "homologous_rel_set n X {} (sum' z \<U>) = \<one>\<^bsub>homology_group p X\<^esub>" for z
|
|
1479 |
proof -
|
|
1480 |
have "z \<in> (\<Pi>\<^sub>E S\<in>\<U>. singular_relcycle_set n (subtopology X S) {})" "finite {S \<in> \<U>. z S \<noteq> 0}"
|
|
1481 |
using z by (auto simp: carrier_sum_group)
|
|
1482 |
have "singular_relboundary n X {} (sum' z \<U>)"
|
|
1483 |
using eq singular_chain_imp_relboundary by (auto simp: relative_homology_group_def peq)
|
|
1484 |
then obtain d where scd: "singular_chain (Suc n) X d" and cbd: "chain_boundary (Suc n) d = sum' z \<U>"
|
|
1485 |
by (auto simp: singular_boundary)
|
|
1486 |
have *: "\<exists>d. singular_chain (Suc n) (subtopology X S) d \<and> chain_boundary (Suc n) d = z S"
|
|
1487 |
if "S \<in> \<U>" for S
|
|
1488 |
proof -
|
|
1489 |
have inj': "inj_on (\<lambda>f. sum' f \<U>) {x \<in> \<Pi>\<^sub>E S\<in>\<U>. singular_chain_set (Suc n) (subtopology X S). finite {S \<in> \<U>. x S \<noteq> 0}}"
|
|
1490 |
using iso_chain_group_sum [OF assms, of "Suc n"]
|
|
1491 |
by (simp add: iso_iff_mon_epi mon_def carrier_sum_group)
|
|
1492 |
obtain w where w: "w \<in> (\<Pi>\<^sub>E S\<in>\<U>. singular_chain_set (Suc n) (subtopology X S))"
|
|
1493 |
and finw: "finite {S \<in> \<U>. w S \<noteq> 0}"
|
|
1494 |
and deq: "d = sum' w \<U>"
|
|
1495 |
using iso_chain_group_sum [OF assms, of "Suc n"] scd
|
|
1496 |
by (auto simp: iso_iff_mon_epi epi_def carrier_sum_group set_eq_iff)
|
|
1497 |
with \<open>S \<in> \<U>\<close> have scwS: "singular_chain (Suc n) (subtopology X S) (w S)"
|
|
1498 |
by blast
|
|
1499 |
have "inj_on (\<lambda>f. sum' f \<U>) {x \<in> \<Pi>\<^sub>E S\<in>\<U>. singular_chain_set n (subtopology X S). finite {S \<in> \<U>. x S \<noteq> 0}}"
|
|
1500 |
using iso_chain_group_sum [OF assms, of n]
|
|
1501 |
by (simp add: iso_iff_mon_epi mon_def carrier_sum_group)
|
|
1502 |
then have "(\<lambda>S\<in>\<U>. chain_boundary (Suc n) (w S)) = z"
|
|
1503 |
proof (rule inj_onD)
|
|
1504 |
have "sum' (\<lambda>S\<in>\<U>. chain_boundary (Suc n) (w S)) \<U> = sum' (chain_boundary (Suc n) \<circ> w) {S \<in> \<U>. w S \<noteq> 0}"
|
|
1505 |
by (auto simp: o_def intro: sum.mono_neutral_right')
|
|
1506 |
also have "\<dots> = chain_boundary (Suc n) d"
|
|
1507 |
by (auto simp: sum.G_def deq chain_boundary_sum finw intro: finite_subset [OF _ finw] sum.mono_neutral_left)
|
|
1508 |
finally show "sum' (\<lambda>S\<in>\<U>. chain_boundary (Suc n) (w S)) \<U> = sum' z \<U>"
|
|
1509 |
by (simp add: cbd)
|
|
1510 |
show "(\<lambda>S\<in>\<U>. chain_boundary (Suc n) (w S)) \<in> {x \<in> \<Pi>\<^sub>E S\<in>\<U>. singular_chain_set n (subtopology X S). finite {S \<in> \<U>. x S \<noteq> 0}}"
|
|
1511 |
using w by (auto simp: PiE_iff singular_chain_boundary_alt cong: rev_conj_cong intro: finite_subset [OF _ finw])
|
|
1512 |
show "z \<in> {x \<in> \<Pi>\<^sub>E S\<in>\<U>. singular_chain_set n (subtopology X S). finite {S \<in> \<U>. x S \<noteq> 0}}"
|
|
1513 |
using z by (simp_all add: carrier_sum_group PiE_iff singular_cycle)
|
|
1514 |
qed
|
|
1515 |
with \<open>S \<in> \<U>\<close> scwS show ?thesis
|
|
1516 |
by force
|
|
1517 |
qed
|
|
1518 |
show ?thesis
|
|
1519 |
apply (rule restrict_ext)
|
|
1520 |
using that *
|
|
1521 |
apply (simp add: singular_boundary relative_homology_group_def homologous_rel_set_eq_relboundary peq)
|
|
1522 |
done
|
|
1523 |
qed
|
|
1524 |
show "x = (\<lambda>S\<in>\<U>. \<one>\<^bsub>homology_group p (subtopology X S)\<^esub>)"
|
|
1525 |
using x 1 carrSG gf
|
|
1526 |
by (auto simp: peq feq)
|
|
1527 |
qed
|
|
1528 |
show "?h ` carrier ?SG = carrier ?HG"
|
|
1529 |
proof safe
|
|
1530 |
fix A
|
|
1531 |
assume "A \<in> carrier (homology_group p X)"
|
|
1532 |
then obtain y where y: "singular_relcycle n X {} y" and xeq: "A = homologous_rel_set n X {} y"
|
|
1533 |
by (auto simp: peq carrier_relative_homology_group)
|
|
1534 |
then obtain x where "x \<in> carrier (sum_group \<U> (\<lambda>T. relcycle_group n (subtopology X T) {}))"
|
|
1535 |
"y = sum' x \<U>"
|
|
1536 |
using iso_cycle_group_sum [OF assms, of n] that by (force simp: iso_iff_mon_epi epi_def)
|
|
1537 |
then show "A \<in> (\<lambda>x. gfinprod (homology_group p X) (\<lambda>V. hom_induced p (subtopology X V) {} X {} id (x V)) \<U>) `
|
|
1538 |
carrier (sum_group \<U> (\<lambda>S. homology_group p (subtopology X S)))"
|
|
1539 |
apply (simp add: carrSG image_comp o_def xeq)
|
|
1540 |
apply (simp add: hom_induced_carrier peq flip: gf cong: gfinprod_cong)
|
|
1541 |
done
|
|
1542 |
qed auto
|
|
1543 |
qed
|
|
1544 |
qed
|
|
1545 |
|
|
1546 |
|
|
1547 |
corollary homology_additivity_axiom:
|
|
1548 |
assumes disj: "pairwise disjnt \<U>" and UU: "\<Union>\<U> = topspace X"
|
|
1549 |
and ope: "\<And>v. v \<in> \<U> \<Longrightarrow> openin X v"
|
|
1550 |
shows "(\<lambda>x. gfinprod (homology_group p X)
|
|
1551 |
(\<lambda>v. hom_induced p (subtopology X v) {} X {} id (x v)) \<U>)
|
|
1552 |
\<in> iso (sum_group \<U> (\<lambda>S. homology_group p (subtopology X S))) (homology_group p X)"
|
|
1553 |
proof (rule homology_additivity_axiom_gen [OF disj UU])
|
|
1554 |
fix C T
|
|
1555 |
assume
|
|
1556 |
"compactin X C" and
|
|
1557 |
"path_connectedin X C" and
|
|
1558 |
"T \<in> \<U>" and
|
|
1559 |
"\<not> disjnt C T"
|
|
1560 |
then have "C \<subseteq> topspace X"
|
|
1561 |
and *: "\<And>B. \<lbrakk>openin X T; T \<inter> B \<inter> C = {}; C \<subseteq> T \<union> B; openin X B\<rbrakk> \<Longrightarrow> B \<inter> C = {}"
|
|
1562 |
apply (auto simp: connectedin disjnt_def dest!: path_connectedin_imp_connectedin, blast)
|
|
1563 |
done
|
|
1564 |
have "C \<subseteq> Union \<U>"
|
|
1565 |
using \<open>C \<subseteq> topspace X\<close> UU by blast
|
|
1566 |
moreover have "\<Union> (\<U> - {T}) \<inter> C = {}"
|
|
1567 |
proof (rule *)
|
|
1568 |
show "T \<inter> \<Union> (\<U> - {T}) \<inter> C = {}"
|
|
1569 |
using \<open>T \<in> \<U>\<close> disj disjointD by fastforce
|
|
1570 |
show "C \<subseteq> T \<union> \<Union> (\<U> - {T})"
|
|
1571 |
using \<open>C \<subseteq> \<Union> \<U>\<close> by fastforce
|
|
1572 |
qed (auto simp: \<open>T \<in> \<U>\<close> ope)
|
|
1573 |
ultimately show "C \<subseteq> T"
|
|
1574 |
by blast
|
|
1575 |
qed
|
|
1576 |
|
|
1577 |
|
|
1578 |
subsection\<open>Special properties of singular homology\<close>
|
|
1579 |
|
|
1580 |
text\<open>In particular: the zeroth homology group is isomorphic to the free abelian group
|
|
1581 |
generated by the path components. So, the "coefficient group" is the integers.\<close>
|
|
1582 |
|
|
1583 |
lemma iso_integer_zeroth_homology_group_aux:
|
|
1584 |
assumes X: "path_connected_space X" and f: "singular_simplex 0 X f" and f': "singular_simplex 0 X f'"
|
|
1585 |
shows "homologous_rel 0 X {} (frag_of f) (frag_of f')"
|
|
1586 |
proof -
|
|
1587 |
let ?p = "\<lambda>j. if j = 0 then 1 else 0"
|
|
1588 |
have "f ?p \<in> topspace X" "f' ?p \<in> topspace X"
|
|
1589 |
using assms by (auto simp: singular_simplex_def continuous_map_def)
|
|
1590 |
then obtain g where g: "pathin X g"
|
|
1591 |
and g0: "g 0 = f ?p"
|
|
1592 |
and g1: "g 1 = f' ?p"
|
|
1593 |
using assms by (force simp: path_connected_space_def)
|
|
1594 |
then have contg: "continuous_map (subtopology euclideanreal {0..1}) X g"
|
|
1595 |
by (simp add: pathin_def)
|
|
1596 |
have "singular_chain (Suc 0) X (frag_of (restrict (g \<circ> (\<lambda>x. x 0)) (standard_simplex 1)))"
|
|
1597 |
proof -
|
|
1598 |
have "continuous_map (subtopology (powertop_real UNIV) (standard_simplex (Suc 0)))
|
|
1599 |
(top_of_set {0..1}) (\<lambda>x. x 0)"
|
|
1600 |
apply (auto simp: continuous_map_in_subtopology g)
|
|
1601 |
apply (metis (mono_tags) UNIV_I continuous_map_from_subtopology continuous_map_product_projection)
|
|
1602 |
apply (simp_all add: standard_simplex_def)
|
|
1603 |
done
|
|
1604 |
moreover have "continuous_map (top_of_set {0..1}) X g"
|
|
1605 |
using contg by blast
|
|
1606 |
ultimately show ?thesis
|
|
1607 |
by (force simp: singular_chain_of chain_boundary_of singular_simplex_def continuous_map_compose)
|
|
1608 |
qed
|
|
1609 |
moreover
|
|
1610 |
have "chain_boundary (Suc 0) (frag_of (restrict (g \<circ> (\<lambda>x. x 0)) (standard_simplex 1))) =
|
|
1611 |
frag_of f - frag_of f'"
|
|
1612 |
proof -
|
|
1613 |
have "singular_face (Suc 0) 0 (g \<circ> (\<lambda>x. x 0)) = f"
|
|
1614 |
"singular_face (Suc 0) (Suc 0) (g \<circ> (\<lambda>x. x 0)) = f'"
|
|
1615 |
using assms
|
|
1616 |
by (auto simp: singular_face_def singular_simplex_def extensional_def simplical_face_def standard_simplex_0 g0 g1)
|
|
1617 |
then show ?thesis
|
|
1618 |
by (simp add: singular_chain_of chain_boundary_of)
|
|
1619 |
qed
|
|
1620 |
ultimately
|
|
1621 |
show ?thesis
|
|
1622 |
by (auto simp: homologous_rel_def singular_boundary)
|
|
1623 |
qed
|
|
1624 |
|
|
1625 |
|
|
1626 |
proposition iso_integer_zeroth_homology_group:
|
|
1627 |
assumes X: "path_connected_space X" and f: "singular_simplex 0 X f"
|
|
1628 |
shows "pow (homology_group 0 X) (homologous_rel_set 0 X {} (frag_of f))
|
|
1629 |
\<in> iso integer_group (homology_group 0 X)" (is "pow ?H ?q \<in> iso _ ?H")
|
|
1630 |
proof -
|
|
1631 |
have srf: "singular_relcycle 0 X {} (frag_of f)"
|
|
1632 |
by (simp add: chain_boundary_def f singular_chain_of singular_cycle)
|
|
1633 |
then have qcarr: "?q \<in> carrier ?H"
|
|
1634 |
by (simp add: carrier_relative_homology_group_0)
|
|
1635 |
have 1: "homologous_rel_set 0 X {} a \<in> range (\<lambda>n. homologous_rel_set 0 X {} (frag_cmul n (frag_of f)))"
|
|
1636 |
if "singular_relcycle 0 X {} a" for a
|
|
1637 |
proof -
|
|
1638 |
have "singular_chain 0 X d \<Longrightarrow>
|
|
1639 |
homologous_rel_set 0 X {} d \<in> range (\<lambda>n. homologous_rel_set 0 X {} (frag_cmul n (frag_of f)))" for d
|
|
1640 |
unfolding singular_chain_def
|
|
1641 |
proof (induction d rule: frag_induction)
|
|
1642 |
case zero
|
|
1643 |
then show ?case
|
|
1644 |
by (metis frag_cmul_zero rangeI)
|
|
1645 |
next
|
|
1646 |
case (one x)
|
|
1647 |
then have "\<exists>i. homologous_rel_set 0 X {} (frag_cmul i (frag_of f))
|
|
1648 |
= homologous_rel_set 0 X {} (frag_of x)"
|
|
1649 |
by (metis (no_types) iso_integer_zeroth_homology_group_aux [OF X] f frag_cmul_one homologous_rel_eq mem_Collect_eq)
|
|
1650 |
with one show ?case
|
|
1651 |
by auto
|
|
1652 |
next
|
|
1653 |
case (diff a b)
|
|
1654 |
then obtain c d where
|
|
1655 |
"homologous_rel 0 X {} (a - b) (frag_cmul c (frag_of f) - frag_cmul d (frag_of f))"
|
|
1656 |
using homologous_rel_diff by (fastforce simp add: homologous_rel_set_eq)
|
|
1657 |
then show ?case
|
|
1658 |
by (rule_tac x="c-d" in image_eqI) (auto simp: homologous_rel_set_eq frag_cmul_diff_distrib)
|
|
1659 |
qed
|
|
1660 |
with that show ?thesis
|
|
1661 |
unfolding singular_relcycle_def by blast
|
|
1662 |
qed
|
|
1663 |
have 2: "n = 0"
|
|
1664 |
if "homologous_rel_set 0 X {} (frag_cmul n (frag_of f)) = \<one>\<^bsub>relative_homology_group 0 X {}\<^esub>"
|
|
1665 |
for n
|
|
1666 |
proof -
|
|
1667 |
have "singular_chain (Suc 0) X d
|
|
1668 |
\<Longrightarrow> frag_extend (\<lambda>x. frag_of f) (chain_boundary (Suc 0) d) = 0" for d
|
|
1669 |
unfolding singular_chain_def
|
|
1670 |
proof (induction d rule: frag_induction)
|
|
1671 |
case (one x)
|
|
1672 |
then show ?case
|
|
1673 |
by (simp add: frag_extend_diff chain_boundary_of)
|
|
1674 |
next
|
|
1675 |
case (diff a b)
|
|
1676 |
then show ?case
|
|
1677 |
by (simp add: chain_boundary_diff frag_extend_diff)
|
|
1678 |
qed auto
|
|
1679 |
with that show ?thesis
|
|
1680 |
by (force simp: singular_boundary relative_homology_group_def homologous_rel_set_eq_relboundary frag_extend_cmul)
|
|
1681 |
qed
|
|
1682 |
interpret GH : group_hom integer_group ?H "([^]\<^bsub>?H\<^esub>) ?q"
|
|
1683 |
by (simp add: group_hom_def group_hom_axioms_def qcarr group.hom_integer_group_pow)
|
|
1684 |
have eq: "pow ?H ?q = (\<lambda>n. homologous_rel_set 0 X {} (frag_cmul n (frag_of f)))"
|
|
1685 |
proof
|
|
1686 |
fix n
|
|
1687 |
have "frag_of f
|
|
1688 |
\<in> carrier (subgroup_generated
|
|
1689 |
(free_Abelian_group (singular_simplex_set 0 X)) (singular_relcycle_set 0 X {}))"
|
|
1690 |
by (metis carrier_relcycle_group chain_group_def mem_Collect_eq relcycle_group_def srf)
|
|
1691 |
then have ff: "frag_of f [^]\<^bsub>relcycle_group 0 X {}\<^esub> n = frag_cmul n (frag_of f)"
|
|
1692 |
by (simp add: relcycle_group_def chain_group_def group.int_pow_subgroup_generated f)
|
|
1693 |
show "pow ?H ?q n = homologous_rel_set 0 X {} (frag_cmul n (frag_of f))"
|
|
1694 |
apply (rule subst [OF right_coset_singular_relboundary])
|
|
1695 |
apply (simp add: relative_homology_group_def)
|
|
1696 |
apply (simp add: srf ff normal.FactGroup_int_pow normal_subgroup_singular_relboundary_relcycle)
|
|
1697 |
done
|
|
1698 |
qed
|
|
1699 |
show ?thesis
|
|
1700 |
apply (subst GH.iso_iff)
|
|
1701 |
apply (simp add: eq)
|
|
1702 |
apply (auto simp: carrier_relative_homology_group_0 1 2)
|
|
1703 |
done
|
|
1704 |
qed
|
|
1705 |
|
|
1706 |
|
|
1707 |
corollary isomorphic_integer_zeroth_homology_group:
|
|
1708 |
assumes X: "path_connected_space X" "topspace X \<noteq> {}"
|
|
1709 |
shows "homology_group 0 X \<cong> integer_group"
|
|
1710 |
proof -
|
|
1711 |
obtain a where a: "a \<in> topspace X"
|
|
1712 |
using assms by auto
|
|
1713 |
have "singular_simplex 0 X (restrict (\<lambda>x. a) (standard_simplex 0))"
|
|
1714 |
by (simp add: singular_simplex_def a)
|
|
1715 |
then show ?thesis
|
|
1716 |
using X group.iso_sym group_integer_group is_isoI iso_integer_zeroth_homology_group by blast
|
|
1717 |
qed
|
|
1718 |
|
|
1719 |
|
|
1720 |
corollary homology_coefficients:
|
|
1721 |
"topspace X = {a} \<Longrightarrow> homology_group 0 X \<cong> integer_group"
|
|
1722 |
using isomorphic_integer_zeroth_homology_group path_connectedin_topspace by fastforce
|
|
1723 |
|
|
1724 |
proposition zeroth_homology_group:
|
|
1725 |
"homology_group 0 X \<cong> free_Abelian_group (path_components_of X)"
|
|
1726 |
proof -
|
|
1727 |
obtain h where h: "h \<in> iso (sum_group (path_components_of X) (\<lambda>S. homology_group 0 (subtopology X S)))
|
|
1728 |
(homology_group 0 X)"
|
|
1729 |
proof (rule that [OF homology_additivity_axiom_gen])
|
|
1730 |
show "disjoint (path_components_of X)"
|
|
1731 |
by (simp add: pairwise_disjoint_path_components_of)
|
|
1732 |
show "\<Union>(path_components_of X) = topspace X"
|
|
1733 |
by (rule Union_path_components_of)
|
|
1734 |
next
|
|
1735 |
fix C T
|
|
1736 |
assume "path_connectedin X C" "T \<in> path_components_of X" "\<not> disjnt C T"
|
|
1737 |
then show "C \<subseteq> T"
|
|
1738 |
by (metis path_components_of_maximal disjnt_sym)+
|
|
1739 |
qed
|
|
1740 |
have "homology_group 0 X \<cong> sum_group (path_components_of X) (\<lambda>S. homology_group 0 (subtopology X S))"
|
|
1741 |
by (rule group.iso_sym) (use h is_iso_def in auto)
|
|
1742 |
also have "\<dots> \<cong> sum_group (path_components_of X) (\<lambda>i. integer_group)"
|
|
1743 |
proof (rule iso_sum_groupI)
|
|
1744 |
show "homology_group 0 (subtopology X i) \<cong> integer_group" if "i \<in> path_components_of X" for i
|
|
1745 |
by (metis that isomorphic_integer_zeroth_homology_group nonempty_path_components_of
|
|
1746 |
path_connectedin_def path_connectedin_path_components_of topspace_subtopology_subset)
|
|
1747 |
qed auto
|
|
1748 |
also have "\<dots> \<cong> free_Abelian_group (path_components_of X)"
|
|
1749 |
using path_connectedin_path_components_of nonempty_path_components_of
|
|
1750 |
by (simp add: isomorphic_sum_integer_group path_connectedin_def)
|
|
1751 |
finally show ?thesis .
|
|
1752 |
qed
|
|
1753 |
|
|
1754 |
|
|
1755 |
lemma isomorphic_homology_imp_path_components:
|
|
1756 |
assumes "homology_group 0 X \<cong> homology_group 0 Y"
|
|
1757 |
shows "path_components_of X \<approx> path_components_of Y"
|
|
1758 |
proof -
|
|
1759 |
have "free_Abelian_group (path_components_of X) \<cong> homology_group 0 X"
|
|
1760 |
by (rule group.iso_sym) (auto simp: zeroth_homology_group)
|
|
1761 |
also have "\<dots> \<cong> homology_group 0 Y"
|
|
1762 |
by (rule assms)
|
|
1763 |
also have "\<dots> \<cong> free_Abelian_group (path_components_of Y)"
|
|
1764 |
by (rule zeroth_homology_group)
|
|
1765 |
finally have "free_Abelian_group (path_components_of X) \<cong> free_Abelian_group (path_components_of Y)" .
|
|
1766 |
then show ?thesis
|
|
1767 |
by (simp add: isomorphic_free_Abelian_groups)
|
|
1768 |
qed
|
|
1769 |
|
|
1770 |
|
|
1771 |
lemma isomorphic_homology_imp_path_connectedness:
|
|
1772 |
assumes "homology_group 0 X \<cong> homology_group 0 Y"
|
|
1773 |
shows "path_connected_space X \<longleftrightarrow> path_connected_space Y"
|
|
1774 |
proof -
|
|
1775 |
obtain h where h: "bij_betw h (path_components_of X) (path_components_of Y)"
|
|
1776 |
using assms isomorphic_homology_imp_path_components eqpoll_def by blast
|
|
1777 |
have 1: "path_components_of X \<subseteq> {a} \<Longrightarrow> path_components_of Y \<subseteq> {h a}" for a
|
|
1778 |
using h unfolding bij_betw_def by blast
|
|
1779 |
have 2: "path_components_of Y \<subseteq> {a}
|
|
1780 |
\<Longrightarrow> path_components_of X \<subseteq> {inv_into (path_components_of X) h a}" for a
|
|
1781 |
using h [THEN bij_betw_inv_into] unfolding bij_betw_def by blast
|
|
1782 |
show ?thesis
|
|
1783 |
unfolding path_connected_space_iff_components_subset_singleton
|
|
1784 |
by (blast intro: dest: 1 2)
|
|
1785 |
qed
|
|
1786 |
|
|
1787 |
|
|
1788 |
subsection\<open>More basic properties of homology groups, deduced from the E-S axioms\<close>
|
|
1789 |
|
|
1790 |
lemma trivial_homology_group:
|
|
1791 |
"p < 0 \<Longrightarrow> trivial_group(homology_group p X)"
|
|
1792 |
by simp
|
|
1793 |
|
|
1794 |
lemma hom_induced_empty_hom:
|
|
1795 |
"(hom_induced p X {} X' {} f) \<in> hom (homology_group p X) (homology_group p X')"
|
|
1796 |
by (simp add: hom_induced_hom)
|
|
1797 |
|
|
1798 |
lemma hom_induced_compose_empty:
|
|
1799 |
"\<lbrakk>continuous_map X Y f; continuous_map Y Z g\<rbrakk>
|
|
1800 |
\<Longrightarrow> hom_induced p X {} Z {} (g \<circ> f) = hom_induced p Y {} Z {} g \<circ> hom_induced p X {} Y {} f"
|
|
1801 |
by (simp add: hom_induced_compose)
|
|
1802 |
|
|
1803 |
lemma homology_homotopy_empty:
|
|
1804 |
"homotopic_with (\<lambda>h. True) X Y f g \<Longrightarrow> hom_induced p X {} Y {} f = hom_induced p X {} Y {} g"
|
|
1805 |
by (simp add: homology_homotopy_axiom)
|
|
1806 |
|
|
1807 |
lemma homotopy_equivalence_relative_homology_group_isomorphisms:
|
|
1808 |
assumes contf: "continuous_map X Y f" and fim: "f ` S \<subseteq> T"
|
|
1809 |
and contg: "continuous_map Y X g" and gim: "g ` T \<subseteq> S"
|
|
1810 |
and gf: "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"
|
|
1811 |
and fg: "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id"
|
|
1812 |
shows "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T)
|
|
1813 |
(hom_induced p X S Y T f) (hom_induced p Y T X S g)"
|
|
1814 |
unfolding group_isomorphisms_def
|
|
1815 |
proof (intro conjI ballI)
|
|
1816 |
fix x
|
|
1817 |
assume x: "x \<in> carrier (relative_homology_group p X S)"
|
|
1818 |
then show "hom_induced p Y T X S g (hom_induced p X S Y T f x) = x"
|
|
1819 |
using homology_homotopy_axiom [OF gf, of p]
|
|
1820 |
apply (simp add: hom_induced_compose [OF contf fim contg gim])
|
|
1821 |
apply (metis comp_apply hom_induced_id)
|
|
1822 |
done
|
|
1823 |
next
|
|
1824 |
fix y
|
|
1825 |
assume "y \<in> carrier (relative_homology_group p Y T)"
|
|
1826 |
then show "hom_induced p X S Y T f (hom_induced p Y T X S g y) = y"
|
|
1827 |
using homology_homotopy_axiom [OF fg, of p]
|
|
1828 |
apply (simp add: hom_induced_compose [OF contg gim contf fim])
|
|
1829 |
apply (metis comp_apply hom_induced_id)
|
|
1830 |
done
|
|
1831 |
qed (auto simp: hom_induced_hom)
|
|
1832 |
|
|
1833 |
|
|
1834 |
lemma homotopy_equivalence_relative_homology_group_isomorphism:
|
|
1835 |
assumes "continuous_map X Y f" and fim: "f ` S \<subseteq> T"
|
|
1836 |
and "continuous_map Y X g" and gim: "g ` T \<subseteq> S"
|
|
1837 |
and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"
|
|
1838 |
and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id"
|
|
1839 |
shows "(hom_induced p X S Y T f) \<in> iso (relative_homology_group p X S) (relative_homology_group p Y T)"
|
|
1840 |
using homotopy_equivalence_relative_homology_group_isomorphisms [OF assms] group_isomorphisms_imp_iso
|
|
1841 |
by metis
|
|
1842 |
|
|
1843 |
lemma homotopy_equivalence_homology_group_isomorphism:
|
|
1844 |
assumes "continuous_map X Y f"
|
|
1845 |
and "continuous_map Y X g"
|
|
1846 |
and "homotopic_with (\<lambda>h. True) X X (g \<circ> f) id"
|
|
1847 |
and "homotopic_with (\<lambda>k. True) Y Y (f \<circ> g) id"
|
|
1848 |
shows "(hom_induced p X {} Y {} f) \<in> iso (homology_group p X) (homology_group p Y)"
|
|
1849 |
apply (rule homotopy_equivalence_relative_homology_group_isomorphism)
|
|
1850 |
using assms by auto
|
|
1851 |
|
|
1852 |
lemma homotopy_equivalent_space_imp_isomorphic_relative_homology_groups:
|
|
1853 |
assumes "continuous_map X Y f" and fim: "f ` S \<subseteq> T"
|
|
1854 |
and "continuous_map Y X g" and gim: "g ` T \<subseteq> S"
|
|
1855 |
and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"
|
|
1856 |
and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id"
|
|
1857 |
shows "relative_homology_group p X S \<cong> relative_homology_group p Y T"
|
|
1858 |
using homotopy_equivalence_relative_homology_group_isomorphism [OF assms]
|
|
1859 |
unfolding is_iso_def by blast
|
|
1860 |
|
|
1861 |
lemma homotopy_equivalent_space_imp_isomorphic_homology_groups:
|
|
1862 |
"X homotopy_equivalent_space Y \<Longrightarrow> homology_group p X \<cong> homology_group p Y"
|
|
1863 |
unfolding homotopy_equivalent_space_def
|
|
1864 |
by (auto intro: homotopy_equivalent_space_imp_isomorphic_relative_homology_groups)
|
|
1865 |
|
|
1866 |
lemma homeomorphic_space_imp_isomorphic_homology_groups:
|
|
1867 |
"X homeomorphic_space Y \<Longrightarrow> homology_group p X \<cong> homology_group p Y"
|
|
1868 |
by (simp add: homeomorphic_imp_homotopy_equivalent_space homotopy_equivalent_space_imp_isomorphic_homology_groups)
|
|
1869 |
|
|
1870 |
lemma trivial_relative_homology_group_gen:
|
|
1871 |
assumes "continuous_map X (subtopology X S) f"
|
|
1872 |
"homotopic_with (\<lambda>h. True) (subtopology X S) (subtopology X S) f id"
|
|
1873 |
"homotopic_with (\<lambda>k. True) X X f id"
|
|
1874 |
shows "trivial_group(relative_homology_group p X S)"
|
|
1875 |
proof (rule exact_seq_imp_triviality)
|
|
1876 |
show "exact_seq ([homology_group (p-1) X,
|
|
1877 |
homology_group (p-1) (subtopology X S),
|
|
1878 |
relative_homology_group p X S, homology_group p X, homology_group p (subtopology X S)],
|
|
1879 |
[hom_induced (p-1) (subtopology X S) {} X {} id,
|
|
1880 |
hom_boundary p X S,
|
|
1881 |
hom_induced p X {} X S id,
|
|
1882 |
hom_induced p (subtopology X S) {} X {} id])"
|
|
1883 |
using homology_exactness_axiom_1 homology_exactness_axiom_2 homology_exactness_axiom_3
|
|
1884 |
by (metis exact_seq_cons_iff)
|
|
1885 |
next
|
|
1886 |
show "hom_induced p (subtopology X S) {} X {} id
|
|
1887 |
\<in> iso (homology_group p (subtopology X S)) (homology_group p X)"
|
|
1888 |
"hom_induced (p - 1) (subtopology X S) {} X {} id
|
|
1889 |
\<in> iso (homology_group (p - 1) (subtopology X S)) (homology_group (p - 1) X)"
|
|
1890 |
using assms
|
|
1891 |
by (auto intro: homotopy_equivalence_relative_homology_group_isomorphism)
|
|
1892 |
qed
|
|
1893 |
|
|
1894 |
|
|
1895 |
lemma trivial_relative_homology_group_topspace:
|
|
1896 |
"trivial_group(relative_homology_group p X (topspace X))"
|
|
1897 |
by (rule trivial_relative_homology_group_gen [where f=id]) auto
|
|
1898 |
|
|
1899 |
lemma trivial_relative_homology_group_empty:
|
|
1900 |
"topspace X = {} \<Longrightarrow> trivial_group(relative_homology_group p X S)"
|
|
1901 |
by (metis Int_absorb2 empty_subsetI relative_homology_group_restrict trivial_relative_homology_group_topspace)
|
|
1902 |
|
|
1903 |
lemma trivial_homology_group_empty:
|
|
1904 |
"topspace X = {} \<Longrightarrow> trivial_group(homology_group p X)"
|
|
1905 |
by (simp add: trivial_relative_homology_group_empty)
|
|
1906 |
|
|
1907 |
lemma homeomorphic_maps_relative_homology_group_isomorphisms:
|
|
1908 |
assumes "homeomorphic_maps X Y f g" and im: "f ` S \<subseteq> T" "g ` T \<subseteq> S"
|
|
1909 |
shows "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T)
|
|
1910 |
(hom_induced p X S Y T f) (hom_induced p Y T X S g)"
|
|
1911 |
proof -
|
|
1912 |
have fg: "continuous_map X Y f" "continuous_map Y X g"
|
|
1913 |
"(\<forall>x\<in>topspace X. g (f x) = x)" "(\<forall>y\<in>topspace Y. f (g y) = y)"
|
|
1914 |
using assms by (simp_all add: homeomorphic_maps_def)
|
|
1915 |
have "group_isomorphisms
|
|
1916 |
(relative_homology_group p X (topspace X \<inter> S))
|
|
1917 |
(relative_homology_group p Y (topspace Y \<inter> T))
|
|
1918 |
(hom_induced p X (topspace X \<inter> S) Y (topspace Y \<inter> T) f)
|
|
1919 |
(hom_induced p Y (topspace Y \<inter> T) X (topspace X \<inter> S) g)"
|
|
1920 |
proof (rule homotopy_equivalence_relative_homology_group_isomorphisms)
|
|
1921 |
show "homotopic_with (\<lambda>h. h ` (topspace X \<inter> S) \<subseteq> topspace X \<inter> S) X X (g \<circ> f) id"
|
|
1922 |
using fg im by (auto intro: homotopic_with_equal continuous_map_compose)
|
|
1923 |
next
|
|
1924 |
show "homotopic_with (\<lambda>k. k ` (topspace Y \<inter> T) \<subseteq> topspace Y \<inter> T) Y Y (f \<circ> g) id"
|
|
1925 |
using fg im by (auto intro: homotopic_with_equal continuous_map_compose)
|
|
1926 |
qed (use im fg in \<open>auto simp: continuous_map_def\<close>)
|
|
1927 |
then show ?thesis
|
|
1928 |
by simp
|
|
1929 |
qed
|
|
1930 |
|
|
1931 |
lemma homeomorphic_map_relative_homology_iso:
|
|
1932 |
assumes f: "homeomorphic_map X Y f" and S: "S \<subseteq> topspace X" "f ` S = T"
|
|
1933 |
shows "(hom_induced p X S Y T f) \<in> iso (relative_homology_group p X S) (relative_homology_group p Y T)"
|
|
1934 |
proof -
|
|
1935 |
obtain g where g: "homeomorphic_maps X Y f g"
|
|
1936 |
using homeomorphic_map_maps f by metis
|
|
1937 |
then have "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T)
|
|
1938 |
(hom_induced p X S Y T f) (hom_induced p Y T X S g)"
|
|
1939 |
using S g by (auto simp: homeomorphic_maps_def intro!: homeomorphic_maps_relative_homology_group_isomorphisms)
|
|
1940 |
then show ?thesis
|
|
1941 |
by (rule group_isomorphisms_imp_iso)
|
|
1942 |
qed
|
|
1943 |
|
|
1944 |
lemma inj_on_hom_induced_section_map:
|
|
1945 |
assumes "section_map X Y f"
|
|
1946 |
shows "inj_on (hom_induced p X {} Y {} f) (carrier (homology_group p X))"
|
|
1947 |
proof -
|
|
1948 |
obtain g where cont: "continuous_map X Y f" "continuous_map Y X g"
|
|
1949 |
and gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x"
|
|
1950 |
using assms by (auto simp: section_map_def retraction_maps_def)
|
|
1951 |
show ?thesis
|
|
1952 |
proof (rule inj_on_inverseI)
|
|
1953 |
fix x
|
|
1954 |
assume x: "x \<in> carrier (homology_group p X)"
|
|
1955 |
have "continuous_map X X (\<lambda>x. g (f x))"
|
|
1956 |
by (metis (no_types, lifting) continuous_map_eq continuous_map_id gf id_apply)
|
|
1957 |
with x show "hom_induced p Y {} X {} g (hom_induced p X {} Y {} f x) = x"
|
|
1958 |
using hom_induced_compose_empty [OF cont, symmetric]
|
|
1959 |
apply (simp add: o_def fun_eq_iff)
|
|
1960 |
apply (rule hom_induced_id_gen)
|
|
1961 |
apply (auto simp: gf)
|
|
1962 |
done
|
|
1963 |
qed
|
|
1964 |
qed
|
|
1965 |
|
|
1966 |
corollary mon_hom_induced_section_map:
|
|
1967 |
assumes "section_map X Y f"
|
|
1968 |
shows "(hom_induced p X {} Y {} f) \<in> mon (homology_group p X) (homology_group p Y)"
|
|
1969 |
by (simp add: hom_induced_empty_hom inj_on_hom_induced_section_map [OF assms] mon_def)
|
|
1970 |
|
|
1971 |
lemma surj_hom_induced_retraction_map:
|
|
1972 |
assumes "retraction_map X Y f"
|
|
1973 |
shows "carrier (homology_group p Y) = (hom_induced p X {} Y {} f) ` carrier (homology_group p X)"
|
|
1974 |
(is "?lhs = ?rhs")
|
|
1975 |
proof -
|
|
1976 |
obtain g where cont: "continuous_map Y X g" "continuous_map X Y f"
|
|
1977 |
and fg: "\<And>x. x \<in> topspace Y \<Longrightarrow> f (g x) = x"
|
|
1978 |
using assms by (auto simp: retraction_map_def retraction_maps_def)
|
|
1979 |
have "x = hom_induced p X {} Y {} f (hom_induced p Y {} X {} g x)"
|
|
1980 |
if x: "x \<in> carrier (homology_group p Y)" for x
|
|
1981 |
proof -
|
|
1982 |
have "continuous_map Y Y (\<lambda>x. f (g x))"
|
|
1983 |
by (metis (no_types, lifting) continuous_map_eq continuous_map_id fg id_apply)
|
|
1984 |
with x show ?thesis
|
|
1985 |
using hom_induced_compose_empty [OF cont, symmetric]
|
|
1986 |
apply (simp add: o_def fun_eq_iff)
|
|
1987 |
apply (rule hom_induced_id_gen [symmetric])
|
|
1988 |
apply (auto simp: fg)
|
|
1989 |
done
|
|
1990 |
qed
|
|
1991 |
moreover
|
|
1992 |
have "(hom_induced p Y {} X {} g x) \<in> carrier (homology_group p X)"
|
|
1993 |
if "x \<in> carrier (homology_group p Y)" for x
|
|
1994 |
by (metis hom_induced)
|
|
1995 |
ultimately have "?lhs \<subseteq> ?rhs"
|
|
1996 |
by auto
|
|
1997 |
moreover have "?rhs \<subseteq> ?lhs"
|
|
1998 |
using hom_induced_hom [of p X "{}" Y "{}" f]
|
|
1999 |
by (simp add: hom_def flip: image_subset_iff_funcset)
|
|
2000 |
ultimately show ?thesis
|
|
2001 |
by auto
|
|
2002 |
qed
|
|
2003 |
|
|
2004 |
|
|
2005 |
corollary epi_hom_induced_retraction_map:
|
|
2006 |
assumes "retraction_map X Y f"
|
|
2007 |
shows "(hom_induced p X {} Y {} f) \<in> epi (homology_group p X) (homology_group p Y)"
|
|
2008 |
using assms epi_iff_subset hom_induced_empty_hom surj_hom_induced_retraction_map by fastforce
|
|
2009 |
|
|
2010 |
|
|
2011 |
lemma homeomorphic_map_homology_iso:
|
|
2012 |
assumes "homeomorphic_map X Y f"
|
|
2013 |
shows "(hom_induced p X {} Y {} f) \<in> iso (homology_group p X) (homology_group p Y)"
|
|
2014 |
using assms
|
|
2015 |
apply (simp add: iso_def bij_betw_def flip: section_and_retraction_eq_homeomorphic_map)
|
|
2016 |
by (metis inj_on_hom_induced_section_map surj_hom_induced_retraction_map hom_induced_hom)
|
|
2017 |
|
|
2018 |
(*See also hom_hom_induced_inclusion*)
|
|
2019 |
lemma inj_on_hom_induced_inclusion:
|
|
2020 |
assumes "S = {} \<or> S retract_of_space X"
|
|
2021 |
shows "inj_on (hom_induced p (subtopology X S) {} X {} id) (carrier (homology_group p (subtopology X S)))"
|
|
2022 |
using assms
|
|
2023 |
proof
|
|
2024 |
assume "S = {}"
|
|
2025 |
then have "trivial_group(homology_group p (subtopology X S))"
|
|
2026 |
by (auto simp: topspace_subtopology intro: trivial_homology_group_empty)
|
|
2027 |
then show ?thesis
|
|
2028 |
by (auto simp: inj_on_def trivial_group_def)
|
|
2029 |
next
|
|
2030 |
assume "S retract_of_space X"
|
|
2031 |
then show ?thesis
|
|
2032 |
by (simp add: retract_of_space_section_map inj_on_hom_induced_section_map)
|
|
2033 |
qed
|
|
2034 |
|
|
2035 |
lemma trivial_homomorphism_hom_boundary_inclusion:
|
|
2036 |
assumes "S = {} \<or> S retract_of_space X"
|
|
2037 |
shows "trivial_homomorphism
|
|
2038 |
(relative_homology_group p X S) (homology_group (p-1) (subtopology X S))
|
|
2039 |
(hom_boundary p X S)"
|
|
2040 |
apply (rule iffD1 [OF exact_seq_mon_eq_triviality inj_on_hom_induced_inclusion [OF assms]])
|
|
2041 |
apply (rule exact_seq.intros)
|
|
2042 |
apply (rule homology_exactness_axiom_1 [of p])
|
|
2043 |
using homology_exactness_axiom_2 [of p]
|
|
2044 |
by auto
|
|
2045 |
|
|
2046 |
lemma epi_hom_induced_relativization:
|
|
2047 |
assumes "S = {} \<or> S retract_of_space X"
|
|
2048 |
shows "(hom_induced p X {} X S id) ` carrier (homology_group p X) = carrier (relative_homology_group p X S)"
|
|
2049 |
apply (rule iffD2 [OF exact_seq_epi_eq_triviality trivial_homomorphism_hom_boundary_inclusion])
|
|
2050 |
apply (rule exact_seq.intros)
|
|
2051 |
apply (rule homology_exactness_axiom_1 [of p])
|
|
2052 |
using homology_exactness_axiom_2 [of p] apply (auto simp: assms)
|
|
2053 |
done
|
|
2054 |
|
|
2055 |
(*different in HOL Light because we don't need short_exact_sequence*)
|
|
2056 |
lemmas short_exact_sequence_hom_induced_inclusion = homology_exactness_axiom_3
|
|
2057 |
|
|
2058 |
lemma group_isomorphisms_homology_group_prod_retract:
|
|
2059 |
assumes "S = {} \<or> S retract_of_space X"
|
|
2060 |
obtains \<H> \<K> where
|
|
2061 |
"subgroup \<H> (homology_group p X)"
|
|
2062 |
"subgroup \<K> (homology_group p X)"
|
|
2063 |
"(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y)
|
|
2064 |
\<in> iso (DirProd (subgroup_generated (homology_group p X) \<H>) (subgroup_generated (homology_group p X) \<K>))
|
|
2065 |
(homology_group p X)"
|
|
2066 |
"(hom_induced p (subtopology X S) {} X {} id)
|
|
2067 |
\<in> iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)"
|
|
2068 |
"(hom_induced p X {} X S id)
|
|
2069 |
\<in> iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)"
|
|
2070 |
using assms
|
|
2071 |
proof
|
|
2072 |
assume "S = {}"
|
|
2073 |
show thesis
|
|
2074 |
proof (rule splitting_lemma_left [OF homology_exactness_axiom_3 [of p]])
|
|
2075 |
let ?f = "\<lambda>x. one(homology_group p (subtopology X {}))"
|
|
2076 |
show "?f \<in> hom (homology_group p X) (homology_group p (subtopology X {}))"
|
|
2077 |
by (simp add: trivial_hom)
|
|
2078 |
have tg: "trivial_group (homology_group p (subtopology X {}))"
|
|
2079 |
by (auto simp: topspace_subtopology trivial_homology_group_empty)
|
|
2080 |
then have [simp]: "carrier (homology_group p (subtopology X {})) = {one (homology_group p (subtopology X {}))}"
|
|
2081 |
by (auto simp: trivial_group_def)
|
|
2082 |
then show "?f (hom_induced p (subtopology X {}) {} X {} id x) = x"
|
|
2083 |
if "x \<in> carrier (homology_group p (subtopology X {}))" for x
|
|
2084 |
using that by auto
|
|
2085 |
show "inj_on (hom_induced p (subtopology X {}) {} X {} id)
|
|
2086 |
(carrier (homology_group p (subtopology X {})))"
|
|
2087 |
by auto
|
|
2088 |
show "hom_induced p X {} X {} id ` carrier (homology_group p X) = carrier (homology_group p X)"
|
|
2089 |
by (metis epi_hom_induced_relativization)
|
|
2090 |
next
|
|
2091 |
fix \<H> \<K>
|
|
2092 |
assume *: "\<H> \<lhd> homology_group p X" "\<K> \<lhd> homology_group p X"
|
|
2093 |
"\<H> \<inter> \<K> \<subseteq> {\<one>\<^bsub>homology_group p X\<^esub>}"
|
|
2094 |
"hom_induced p (subtopology X {}) {} X {} id
|
|
2095 |
\<in> Group.iso (homology_group p (subtopology X {})) (subgroup_generated (homology_group p X) \<H>)"
|
|
2096 |
"hom_induced p X {} X {} id
|
|
2097 |
\<in> Group.iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X {})"
|
|
2098 |
"\<H> <#>\<^bsub>homology_group p X\<^esub> \<K> = carrier (homology_group p X)"
|
|
2099 |
show thesis
|
|
2100 |
proof (rule that)
|
|
2101 |
show "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y)
|
|
2102 |
\<in> iso (subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>)
|
|
2103 |
(homology_group p X)"
|
|
2104 |
using * by (simp add: group_disjoint_sum.iso_group_mul normal_def group_disjoint_sum_def)
|
|
2105 |
qed (use \<open>S = {}\<close> * in \<open>auto simp: normal_def\<close>)
|
|
2106 |
qed
|
|
2107 |
next
|
|
2108 |
assume "S retract_of_space X"
|
|
2109 |
then obtain r where "S \<subseteq> topspace X" and r: "continuous_map X (subtopology X S) r"
|
|
2110 |
and req: "\<forall>x \<in> S. r x = x"
|
|
2111 |
by (auto simp: retract_of_space_def)
|
|
2112 |
show thesis
|
|
2113 |
proof (rule splitting_lemma_left [OF homology_exactness_axiom_3 [of p]])
|
|
2114 |
let ?f = "hom_induced p X {} (subtopology X S) {} r"
|
|
2115 |
show "?f \<in> hom (homology_group p X) (homology_group p (subtopology X S))"
|
|
2116 |
by (simp add: hom_induced_empty_hom)
|
|
2117 |
show eqx: "?f (hom_induced p (subtopology X S) {} X {} id x) = x"
|
|
2118 |
if "x \<in> carrier (homology_group p (subtopology X S))" for x
|
|
2119 |
proof -
|
|
2120 |
have "hom_induced p (subtopology X S) {} (subtopology X S) {} r x = x"
|
|
2121 |
by (metis \<open>S \<subseteq> topspace X\<close> continuous_map_from_subtopology hom_induced_id_gen inf.absorb_iff2 r req that topspace_subtopology)
|
|
2122 |
then show ?thesis
|
|
2123 |
by (simp add: r hom_induced_compose [unfolded o_def fun_eq_iff, rule_format, symmetric])
|
|
2124 |
qed
|
|
2125 |
then show "inj_on (hom_induced p (subtopology X S) {} X {} id)
|
|
2126 |
(carrier (homology_group p (subtopology X S)))"
|
|
2127 |
unfolding inj_on_def by metis
|
|
2128 |
show "hom_induced p X {} X S id ` carrier (homology_group p X) = carrier (relative_homology_group p X S)"
|
|
2129 |
by (simp add: \<open>S retract_of_space X\<close> epi_hom_induced_relativization)
|
|
2130 |
next
|
|
2131 |
fix \<H> \<K>
|
|
2132 |
assume *: "\<H> \<lhd> homology_group p X" "\<K> \<lhd> homology_group p X"
|
|
2133 |
"\<H> \<inter> \<K> \<subseteq> {\<one>\<^bsub>homology_group p X\<^esub>}"
|
|
2134 |
"\<H> <#>\<^bsub>homology_group p X\<^esub> \<K> = carrier (homology_group p X)"
|
|
2135 |
"hom_induced p (subtopology X S) {} X {} id
|
|
2136 |
\<in> Group.iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)"
|
|
2137 |
"hom_induced p X {} X S id
|
|
2138 |
\<in> Group.iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)"
|
|
2139 |
show "thesis"
|
|
2140 |
proof (rule that)
|
|
2141 |
show "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y)
|
|
2142 |
\<in> iso (subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>)
|
|
2143 |
(homology_group p X)"
|
|
2144 |
using *
|
|
2145 |
by (simp add: group_disjoint_sum.iso_group_mul normal_def group_disjoint_sum_def)
|
|
2146 |
qed (use * in \<open>auto simp: normal_def\<close>)
|
|
2147 |
qed
|
|
2148 |
qed
|
|
2149 |
|
|
2150 |
|
|
2151 |
|
|
2152 |
lemma isomorphic_group_homology_group_prod_retract:
|
|
2153 |
assumes "S = {} \<or> S retract_of_space X"
|
|
2154 |
shows "homology_group p X \<cong> homology_group p (subtopology X S) \<times>\<times> relative_homology_group p X S"
|
|
2155 |
(is "?lhs \<cong> ?rhs")
|
|
2156 |
proof -
|
|
2157 |
obtain \<H> \<K> where
|
|
2158 |
"subgroup \<H> (homology_group p X)"
|
|
2159 |
"subgroup \<K> (homology_group p X)"
|
|
2160 |
and 1: "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y)
|
|
2161 |
\<in> iso (DirProd (subgroup_generated (homology_group p X) \<H>) (subgroup_generated (homology_group p X) \<K>))
|
|
2162 |
(homology_group p X)"
|
|
2163 |
"(hom_induced p (subtopology X S) {} X {} id)
|
|
2164 |
\<in> iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)"
|
|
2165 |
"(hom_induced p X {} X S id)
|
|
2166 |
\<in> iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)"
|
|
2167 |
using group_isomorphisms_homology_group_prod_retract [OF assms] by blast
|
|
2168 |
have "?lhs \<cong> subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>"
|
|
2169 |
by (meson DirProd_group 1 abelian_homology_group comm_group_def group.abelian_subgroup_generated group.iso_sym is_isoI)
|
|
2170 |
also have "\<dots> \<cong> ?rhs"
|
|
2171 |
by (meson "1"(2) "1"(3) abelian_homology_group comm_group_def group.DirProd_iso_trans group.abelian_subgroup_generated group.iso_sym is_isoI)
|
|
2172 |
finally show ?thesis .
|
|
2173 |
qed
|
|
2174 |
|
|
2175 |
|
|
2176 |
lemma homology_additivity_explicit:
|
|
2177 |
assumes "openin X S" "openin X T" "disjnt S T" and SUT: "S \<union> T = topspace X"
|
|
2178 |
shows "(\<lambda>(a,b).(hom_induced p (subtopology X S) {} X {} id a)
|
|
2179 |
\<otimes>\<^bsub>homology_group p X\<^esub>
|
|
2180 |
(hom_induced p (subtopology X T) {} X {} id b))
|
|
2181 |
\<in> iso (DirProd (homology_group p (subtopology X S)) (homology_group p (subtopology X T)))
|
|
2182 |
(homology_group p X)"
|
|
2183 |
proof -
|
|
2184 |
have "closedin X S" "closedin X T"
|
|
2185 |
using assms Un_commute disjnt_sym
|
|
2186 |
by (metis Diff_cancel Diff_triv Un_Diff disjnt_def openin_closedin_eq sup_bot.right_neutral)+
|
|
2187 |
with \<open>openin X S\<close> \<open>openin X T\<close> have SS: "X closure_of S \<subseteq> X interior_of S" and TT: "X closure_of T \<subseteq> X interior_of T"
|
|
2188 |
by (simp_all add: closure_of_closedin interior_of_openin)
|
|
2189 |
have [simp]: "S \<union> T - T = S" "S \<union> T - S = T"
|
|
2190 |
using \<open>disjnt S T\<close>
|
|
2191 |
by (auto simp: Diff_triv Un_Diff disjnt_def)
|
|
2192 |
let ?f = "hom_induced p X {} X T id"
|
|
2193 |
let ?g = "hom_induced p X {} X S id"
|
|
2194 |
let ?h = "hom_induced p (subtopology X S) {} X T id"
|
|
2195 |
let ?i = "hom_induced p (subtopology X S) {} X {} id"
|
|
2196 |
let ?j = "hom_induced p (subtopology X T) {} X {} id"
|
|
2197 |
let ?k = "hom_induced p (subtopology X T) {} X S id"
|
|
2198 |
let ?A = "homology_group p (subtopology X S)"
|
|
2199 |
let ?B = "homology_group p (subtopology X T)"
|
|
2200 |
let ?C = "relative_homology_group p X T"
|
|
2201 |
let ?D = "relative_homology_group p X S"
|
|
2202 |
let ?G = "homology_group p X"
|
|
2203 |
have h: "?h \<in> iso ?A ?C" and k: "?k \<in> iso ?B ?D"
|
|
2204 |
using homology_excision_axiom [OF TT, of "S \<union> T" p]
|
|
2205 |
using homology_excision_axiom [OF SS, of "S \<union> T" p]
|
|
2206 |
by auto (simp_all add: SUT)
|
|
2207 |
have 1: "\<And>x. (hom_induced p X {} X T id \<circ> hom_induced p (subtopology X S) {} X {} id) x
|
|
2208 |
= hom_induced p (subtopology X S) {} X T id x"
|
|
2209 |
by (simp flip: hom_induced_compose)
|
|
2210 |
have 2: "\<And>x. (hom_induced p X {} X S id \<circ> hom_induced p (subtopology X T) {} X {} id) x
|
|
2211 |
= hom_induced p (subtopology X T) {} X S id x"
|
|
2212 |
by (simp flip: hom_induced_compose)
|
|
2213 |
show ?thesis
|
|
2214 |
using exact_sequence_sum_lemma
|
|
2215 |
[OF abelian_homology_group h k homology_exactness_axiom_3 homology_exactness_axiom_3] 1 2
|
|
2216 |
by auto
|
|
2217 |
qed
|
|
2218 |
|
|
2219 |
|
|
2220 |
subsection\<open>Generalize exact homology sequence to triples\<close>
|
|
2221 |
|
|
2222 |
definition hom_relboundary :: "[int,'a topology,'a set,'a set,'a chain set] \<Rightarrow> 'a chain set"
|
|
2223 |
where
|
|
2224 |
"hom_relboundary p X S T =
|
|
2225 |
hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id \<circ>
|
|
2226 |
hom_boundary p X S"
|
|
2227 |
|
|
2228 |
lemma group_homomorphism_hom_relboundary:
|
|
2229 |
"hom_relboundary p X S T
|
|
2230 |
\<in> hom (relative_homology_group p X S) (relative_homology_group (p - 1) (subtopology X S) T)"
|
|
2231 |
unfolding hom_relboundary_def
|
|
2232 |
proof (rule hom_compose)
|
|
2233 |
show "hom_boundary p X S \<in> hom (relative_homology_group p X S) (homology_group(p - 1) (subtopology X S))"
|
|
2234 |
by (simp add: hom_boundary_hom)
|
|
2235 |
show "hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id
|
|
2236 |
\<in> hom (homology_group(p - 1) (subtopology X S)) (relative_homology_group (p - 1) (subtopology X S) T)"
|
|
2237 |
by (simp add: hom_induced_hom)
|
|
2238 |
qed
|
|
2239 |
|
|
2240 |
lemma hom_relboundary:
|
|
2241 |
"hom_relboundary p X S T c \<in> carrier (relative_homology_group (p - 1) (subtopology X S) T)"
|
|
2242 |
by (simp add: hom_relboundary_def hom_induced_carrier)
|
|
2243 |
|
|
2244 |
lemma hom_relboundary_empty: "hom_relboundary p X S {} = hom_boundary p X S"
|
|
2245 |
apply (simp add: hom_relboundary_def o_def)
|
|
2246 |
apply (subst hom_induced_id)
|
|
2247 |
apply (metis hom_boundary_carrier, auto)
|
|
2248 |
done
|
|
2249 |
|
|
2250 |
lemma naturality_hom_induced_relboundary:
|
|
2251 |
assumes "continuous_map X Y f" "f ` S \<subseteq> U" "f ` T \<subseteq> V"
|
|
2252 |
shows "hom_relboundary p Y U V \<circ>
|
|
2253 |
hom_induced p X S Y (U) f =
|
|
2254 |
hom_induced (p - 1) (subtopology X S) T (subtopology Y U) V f \<circ>
|
|
2255 |
hom_relboundary p X S T"
|
|
2256 |
proof -
|
|
2257 |
have [simp]: "continuous_map (subtopology X S) (subtopology Y U) f"
|
|
2258 |
using assms continuous_map_from_subtopology continuous_map_in_subtopology topspace_subtopology by fastforce
|
|
2259 |
have "hom_induced (p - 1) (subtopology Y U) {} (subtopology Y U) V id \<circ>
|
|
2260 |
hom_induced (p - 1) (subtopology X S) {} (subtopology Y U) {} f
|
|
2261 |
= hom_induced (p - 1) (subtopology X S) T (subtopology Y U) V f \<circ>
|
|
2262 |
hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id"
|
|
2263 |
using assms by (simp flip: hom_induced_compose)
|
|
2264 |
then show ?thesis
|
|
2265 |
apply (simp add: hom_relboundary_def comp_assoc naturality_hom_induced assms)
|
|
2266 |
apply (simp flip: comp_assoc)
|
|
2267 |
done
|
|
2268 |
qed
|
|
2269 |
|
|
2270 |
proposition homology_exactness_triple_1:
|
|
2271 |
assumes "T \<subseteq> S"
|
|
2272 |
shows "exact_seq ([relative_homology_group(p - 1) (subtopology X S) T,
|
|
2273 |
relative_homology_group p X S,
|
|
2274 |
relative_homology_group p X T],
|
|
2275 |
[hom_relboundary p X S T, hom_induced p X T X S id])"
|
|
2276 |
(is "exact_seq ([?G1,?G2,?G3], [?h1,?h2])")
|
|
2277 |
proof -
|
|
2278 |
have iTS: "id ` T \<subseteq> S" and [simp]: "S \<inter> T = T"
|
|
2279 |
using assms by auto
|
|
2280 |
have "?h2 B \<in> kernel ?G2 ?G1 ?h1" for B
|
|
2281 |
proof -
|
|
2282 |
have "hom_boundary p X T B \<in> carrier (relative_homology_group (p - 1) (subtopology X T) {})"
|
|
2283 |
by (metis (no_types) hom_boundary)
|
|
2284 |
then have *: "hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id
|
|
2285 |
(hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id
|
|
2286 |
(hom_boundary p X T B))
|
|
2287 |
= \<one>\<^bsub>?G1\<^esub>"
|
|
2288 |
using homology_exactness_axiom_3 [of "p-1" "subtopology X S" T]
|
|
2289 |
by (auto simp: subtopology_subtopology kernel_def)
|
|
2290 |
show ?thesis
|
|
2291 |
apply (simp add: kernel_def hom_induced_carrier hom_relboundary_def flip: *)
|
|
2292 |
by (metis comp_def naturality_hom_induced [OF continuous_map_id iTS])
|
|
2293 |
qed
|
|
2294 |
moreover have "B \<in> ?h2 ` carrier ?G3" if "B \<in> kernel ?G2 ?G1 ?h1" for B
|
|
2295 |
proof -
|
|
2296 |
have Bcarr: "B \<in> carrier ?G2"
|
|
2297 |
and Beq: "?h1 B = \<one>\<^bsub>?G1\<^esub>"
|
|
2298 |
using that by (auto simp: kernel_def)
|
|
2299 |
have "\<exists>A' \<in> carrier (homology_group (p - 1) (subtopology X T)). hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id A' = A"
|
|
2300 |
if "A \<in> carrier (homology_group (p - 1) (subtopology X S))"
|
|
2301 |
"hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id A =
|
|
2302 |
\<one>\<^bsub>?G1\<^esub>" for A
|
|
2303 |
using homology_exactness_axiom_3 [of "p-1" "subtopology X S" T] that
|
|
2304 |
by (simp add: kernel_def subtopology_subtopology image_iff set_eq_iff) meson
|
|
2305 |
then obtain C where Ccarr: "C \<in> carrier (homology_group (p - 1) (subtopology X T))"
|
|
2306 |
and Ceq: "hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id C = hom_boundary p X S B"
|
|
2307 |
using Beq by (simp add: hom_relboundary_def) (metis hom_boundary_carrier)
|
|
2308 |
let ?hi_XT = "hom_induced (p - 1) (subtopology X T) {} X {} id"
|
|
2309 |
have "?hi_XT
|
|
2310 |
= hom_induced (p - 1) (subtopology X S) {} X {} id
|
|
2311 |
\<circ> (hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id)"
|
|
2312 |
by (metis assms comp_id continuous_map_id_subt hom_induced_compose_empty inf.absorb_iff2 subtopology_subtopology)
|
|
2313 |
then have "?hi_XT C
|
|
2314 |
= hom_induced (p - 1) (subtopology X S) {} X {} id (hom_boundary p X S B)"
|
|
2315 |
by (simp add: Ceq)
|
|
2316 |
also have eq: "\<dots> = \<one>\<^bsub>homology_group (p - 1) X\<^esub>"
|
|
2317 |
using homology_exactness_axiom_2 [of p X S] Bcarr by (auto simp: kernel_def)
|
|
2318 |
finally have "?hi_XT C = \<one>\<^bsub>homology_group (p - 1) X\<^esub>" .
|
|
2319 |
then obtain D where Dcarr: "D \<in> carrier ?G3" and Deq: "hom_boundary p X T D = C"
|
|
2320 |
using homology_exactness_axiom_2 [of p X T] Ccarr
|
|
2321 |
by (auto simp: kernel_def image_iff set_eq_iff) meson
|
|
2322 |
interpret hb: group_hom "?G2" "homology_group (p-1) (subtopology X S)"
|
|
2323 |
"hom_boundary p X S"
|
|
2324 |
using hom_boundary_hom group_hom_axioms_def group_hom_def by fastforce
|
|
2325 |
let ?A = "B \<otimes>\<^bsub>?G2\<^esub> inv\<^bsub>?G2\<^esub> ?h2 D"
|
|
2326 |
have "\<exists>A' \<in> carrier (homology_group p X). hom_induced p X {} X S id A' = A"
|
|
2327 |
if "A \<in> carrier ?G2"
|
|
2328 |
"hom_boundary p X S A = one (homology_group (p - 1) (subtopology X S))" for A
|
|
2329 |
using that homology_exactness_axiom_1 [of p X S]
|
|
2330 |
by (simp add: kernel_def subtopology_subtopology image_iff set_eq_iff) meson
|
|
2331 |
moreover
|
|
2332 |
have "?A \<in> carrier ?G2"
|
|
2333 |
by (simp add: Bcarr abelian_relative_homology_group comm_groupE(1) hom_induced_carrier)
|
|
2334 |
moreover have "hom_boundary p X S (?h2 D) = hom_boundary p X S B"
|
|
2335 |
by (metis (mono_tags, lifting) Ceq Deq comp_eq_dest continuous_map_id iTS naturality_hom_induced)
|
|
2336 |
then have "hom_boundary p X S ?A = one (homology_group (p - 1) (subtopology X S))"
|
|
2337 |
by (simp add: hom_induced_carrier Bcarr)
|
|
2338 |
ultimately obtain W where Wcarr: "W \<in> carrier (homology_group p X)"
|
|
2339 |
and Weq: "hom_induced p X {} X S id W = ?A"
|
|
2340 |
by blast
|
|
2341 |
let ?W = "D \<otimes>\<^bsub>?G3\<^esub> hom_induced p X {} X T id W"
|
|
2342 |
show ?thesis
|
|
2343 |
proof
|
|
2344 |
interpret comm_group "?G2"
|
|
2345 |
by (rule abelian_relative_homology_group)
|
|
2346 |
have "B = (?h2 \<circ> hom_induced p X {} X T id) W \<otimes>\<^bsub>?G2\<^esub> ?h2 D"
|
|
2347 |
apply (simp add: hom_induced_compose [symmetric] assms)
|
|
2348 |
by (metis Bcarr Weq hb.G.inv_solve_right hom_induced_carrier)
|
|
2349 |
then have "B \<otimes>\<^bsub>?G2\<^esub> inv\<^bsub>?G2\<^esub> ?h2 D
|
|
2350 |
= ?h2 (hom_induced p X {} X T id W)"
|
|
2351 |
by (simp add: hb.G.m_assoc hom_induced_carrier)
|
|
2352 |
then show "B = ?h2 ?W"
|
|
2353 |
apply (simp add: Dcarr hom_induced_carrier hom_mult [OF hom_induced_hom])
|
|
2354 |
by (metis Bcarr hb.G.inv_solve_right hom_induced_carrier m_comm)
|
|
2355 |
show "?W \<in> carrier ?G3"
|
|
2356 |
by (simp add: Dcarr abelian_relative_homology_group comm_groupE(1) hom_induced_carrier)
|
|
2357 |
qed
|
|
2358 |
qed
|
|
2359 |
ultimately show ?thesis
|
|
2360 |
by (auto simp: group_hom_def group_hom_axioms_def hom_induced_hom group_homomorphism_hom_relboundary)
|
|
2361 |
qed
|
|
2362 |
|
|
2363 |
|
|
2364 |
proposition homology_exactness_triple_2:
|
|
2365 |
assumes "T \<subseteq> S"
|
|
2366 |
shows "exact_seq ([relative_homology_group(p - 1) X T,
|
|
2367 |
relative_homology_group(p - 1) (subtopology X S) T,
|
|
2368 |
relative_homology_group p X S],
|
|
2369 |
[hom_induced (p - 1) (subtopology X S) T X T id, hom_relboundary p X S T])"
|
|
2370 |
(is "exact_seq ([?G1,?G2,?G3], [?h1,?h2])")
|
|
2371 |
proof -
|
|
2372 |
let ?H2 = "homology_group (p - 1) (subtopology X S)"
|
|
2373 |
have iTS: "id ` T \<subseteq> S" and [simp]: "S \<inter> T = T"
|
|
2374 |
using assms by auto
|
|
2375 |
have "?h2 C \<in> kernel ?G2 ?G1 ?h1" for C
|
|
2376 |
proof -
|
|
2377 |
have "?h1 (?h2 C)
|
|
2378 |
= (hom_induced (p - 1) X {} X T id \<circ> hom_induced (p - 1) (subtopology X S) {} X {} id \<circ> hom_boundary p X S) C"
|
|
2379 |
unfolding hom_relboundary_def
|
|
2380 |
by (metis (no_types, lifting) comp_apply continuous_map_id continuous_map_id_subt empty_subsetI hom_induced_compose id_apply image_empty image_id order_refl)
|
|
2381 |
also have "\<dots> = \<one>\<^bsub>?G1\<^esub>"
|
|
2382 |
proof -
|
|
2383 |
have *: "hom_boundary p X S C \<in> carrier ?H2"
|
|
2384 |
by (simp add: hom_boundary_carrier)
|
|
2385 |
moreover have "hom_boundary p X S C \<in> hom_boundary p X S ` carrier ?G3"
|
|
2386 |
using homology_exactness_axiom_2 [of p X S] *
|
|
2387 |
apply (simp add: kernel_def set_eq_iff)
|
|
2388 |
by (metis group_relative_homology_group hom_boundary_default hom_one image_eqI)
|
|
2389 |
ultimately
|
|
2390 |
have 1: "hom_induced (p - 1) (subtopology X S) {} X {} id (hom_boundary p X S C)
|
|
2391 |
= \<one>\<^bsub>homology_group (p - 1) X\<^esub>"
|
|
2392 |
using homology_exactness_axiom_2 [of p X S] by (simp add: kernel_def) blast
|
|
2393 |
show ?thesis
|
|
2394 |
by (simp add: 1 hom_one [OF hom_induced_hom])
|
|
2395 |
qed
|
|
2396 |
finally have "?h1 (?h2 C) = \<one>\<^bsub>?G1\<^esub>" .
|
|
2397 |
then show ?thesis
|
|
2398 |
by (simp add: kernel_def hom_relboundary_def hom_induced_carrier)
|
|
2399 |
qed
|
|
2400 |
moreover have "x \<in> ?h2 ` carrier ?G3" if "x \<in> kernel ?G2 ?G1 ?h1" for x
|
|
2401 |
proof -
|
|
2402 |
let ?homX = "hom_induced (p - 1) (subtopology X S) {} X {} id"
|
|
2403 |
let ?homXS = "hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id"
|
|
2404 |
have "x \<in> carrier (relative_homology_group (p - 1) (subtopology X S) T)"
|
|
2405 |
using that by (simp add: kernel_def)
|
|
2406 |
moreover
|
|
2407 |
have "hom_boundary (p-1) X T \<circ> hom_induced (p-1) (subtopology X S) T X T id = hom_boundary (p-1) (subtopology X S) T"
|
|
2408 |
by (metis Int_lower2 \<open>S \<inter> T = T\<close> continuous_map_id_subt hom_relboundary_def hom_relboundary_empty id_apply image_id naturality_hom_induced subtopology_subtopology)
|
|
2409 |
then have "hom_boundary (p - 1) (subtopology X S) T x = \<one>\<^bsub>homology_group (p - 2) (subtopology (subtopology X S) T)\<^esub>"
|
|
2410 |
using naturality_hom_induced [of "subtopology X S" X id T T "p-1"] that
|
|
2411 |
hom_one [OF hom_boundary_hom group_relative_homology_group group_relative_homology_group, of "p-1" X T]
|
|
2412 |
apply (simp add: kernel_def subtopology_subtopology)
|
|
2413 |
by (metis comp_apply)
|
|
2414 |
ultimately
|
|
2415 |
obtain y where ycarr: "y \<in> carrier ?H2"
|
|
2416 |
and yeq: "?homXS y = x"
|
|
2417 |
using homology_exactness_axiom_1 [of "p-1" "subtopology X S" T]
|
|
2418 |
by (simp add: kernel_def image_def set_eq_iff) meson
|
|
2419 |
have "?homX y \<in> carrier (homology_group (p - 1) X)"
|
|
2420 |
by (simp add: hom_induced_carrier)
|
|
2421 |
moreover
|
|
2422 |
have "(hom_induced (p - 1) X {} X T id \<circ> ?homX) y = \<one>\<^bsub>relative_homology_group (p - 1) X T\<^esub>"
|
|
2423 |
apply (simp flip: hom_induced_compose)
|
|
2424 |
using hom_induced_compose [of "subtopology X S" "subtopology X S" id "{}" T X id T "p-1"]
|
|
2425 |
apply simp
|
|
2426 |
by (metis (mono_tags, lifting) kernel_def mem_Collect_eq that yeq)
|
|
2427 |
then have "hom_induced (p - 1) X {} X T id (?homX y) = \<one>\<^bsub>relative_homology_group (p - 1) X T\<^esub>"
|
|
2428 |
by simp
|
|
2429 |
ultimately obtain z where zcarr: "z \<in> carrier (homology_group (p - 1) (subtopology X T))"
|
|
2430 |
and zeq: "hom_induced (p - 1) (subtopology X T) {} X {} id z = ?homX y"
|
|
2431 |
using homology_exactness_axiom_3 [of "p-1" X T]
|
|
2432 |
by (auto simp: kernel_def dest!: equalityD1 [of "Collect _"])
|
|
2433 |
have *: "\<And>t. \<lbrakk>t \<in> carrier ?H2;
|
|
2434 |
hom_induced (p - 1) (subtopology X S) {} X {} id t = \<one>\<^bsub>homology_group (p - 1) X\<^esub>\<rbrakk>
|
|
2435 |
\<Longrightarrow> t \<in> hom_boundary p X S ` carrier ?G3"
|
|
2436 |
using homology_exactness_axiom_2 [of p X S]
|
|
2437 |
by (auto simp: kernel_def dest!: equalityD1 [of "Collect _"])
|
|
2438 |
interpret comm_group "?H2"
|
|
2439 |
by (rule abelian_relative_homology_group)
|
|
2440 |
interpret gh: group_hom ?H2 "homology_group (p - 1) X" "hom_induced (p-1) (subtopology X S) {} X {} id"
|
|
2441 |
by (meson group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced)
|
|
2442 |
let ?yz = "y \<otimes>\<^bsub>?H2\<^esub> inv\<^bsub>?H2\<^esub> hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id z"
|
|
2443 |
have yzcarr: "?yz \<in> carrier ?H2"
|
|
2444 |
by (simp add: hom_induced_carrier ycarr)
|
|
2445 |
have yzeq: "hom_induced (p - 1) (subtopology X S) {} X {} id ?yz = \<one>\<^bsub>homology_group (p - 1) X\<^esub>"
|
|
2446 |
apply (simp add: hom_induced_carrier ycarr gh.inv_solve_right')
|
|
2447 |
by (metis assms continuous_map_id_subt hom_induced_compose_empty inf.absorb_iff2 o_apply o_id subtopology_subtopology zeq)
|
|
2448 |
obtain w where wcarr: "w \<in> carrier ?G3" and weq: "hom_boundary p X S w = ?yz"
|
|
2449 |
using * [OF yzcarr yzeq] by blast
|
|
2450 |
interpret gh2: group_hom ?H2 ?G2 ?homXS
|
|
2451 |
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
|
|
2452 |
have "?homXS (hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id z)
|
|
2453 |
= \<one>\<^bsub>relative_homology_group (p - 1) (subtopology X S) T\<^esub>"
|
|
2454 |
using homology_exactness_axiom_3 [of "p-1" "subtopology X S" T] zcarr
|
|
2455 |
by (auto simp: kernel_def subtopology_subtopology)
|
|
2456 |
then show ?thesis
|
|
2457 |
apply (rule_tac x=w in image_eqI)
|
|
2458 |
apply (simp_all add: hom_relboundary_def weq wcarr)
|
|
2459 |
by (metis gh2.hom_inv gh2.hom_mult gh2.inv_one gh2.r_one group.inv_closed group_l_invI hom_induced_carrier l_inv_ex ycarr yeq)
|
|
2460 |
qed
|
|
2461 |
ultimately show ?thesis
|
|
2462 |
by (auto simp: group_hom_axioms_def group_hom_def group_homomorphism_hom_relboundary hom_induced_hom)
|
|
2463 |
qed
|
|
2464 |
|
|
2465 |
proposition homology_exactness_triple_3:
|
|
2466 |
assumes "T \<subseteq> S"
|
|
2467 |
shows "exact_seq ([relative_homology_group p X S,
|
|
2468 |
relative_homology_group p X T,
|
|
2469 |
relative_homology_group p (subtopology X S) T],
|
|
2470 |
[hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])"
|
|
2471 |
(is "exact_seq ([?G1,?G2,?G3], [?h1,?h2])")
|
|
2472 |
proof -
|
|
2473 |
have iTS: "id ` T \<subseteq> S" and [simp]: "S \<inter> T = T"
|
|
2474 |
using assms by auto
|
|
2475 |
have 1: "?h2 x \<in> kernel ?G2 ?G1 ?h1" for x
|
|
2476 |
proof -
|
|
2477 |
have "?h1 (?h2 x)
|
|
2478 |
= (hom_induced p (subtopology X S) S X S id \<circ>
|
|
2479 |
hom_induced p (subtopology X S) T (subtopology X S) S id) x"
|
|
2480 |
by (metis comp_eq_dest_lhs continuous_map_id continuous_map_id_subt hom_induced_compose iTS id_apply image_subsetI)
|
|
2481 |
also have "\<dots> = \<one>\<^bsub>relative_homology_group p X S\<^esub>"
|
|
2482 |
proof -
|
|
2483 |
have "trivial_group (relative_homology_group p (subtopology X S) S)"
|
|
2484 |
using trivial_relative_homology_group_topspace [of p "subtopology X S"]
|
|
2485 |
by (metis inf_right_idem relative_homology_group_restrict topspace_subtopology)
|
|
2486 |
then have 1: "hom_induced p (subtopology X S) T (subtopology X S) S id x
|
|
2487 |
= \<one>\<^bsub>relative_homology_group p (subtopology X S) S\<^esub>"
|
|
2488 |
using hom_induced_carrier by (fastforce simp add: trivial_group_def)
|
|
2489 |
show ?thesis
|
|
2490 |
by (simp add: 1 hom_one [OF hom_induced_hom])
|
|
2491 |
qed
|
|
2492 |
finally have "?h1 (?h2 x) = \<one>\<^bsub>relative_homology_group p X S\<^esub>" .
|
|
2493 |
then show ?thesis
|
|
2494 |
by (simp add: hom_induced_carrier kernel_def)
|
|
2495 |
qed
|
|
2496 |
moreover have "x \<in> ?h2 ` carrier ?G3" if x: "x \<in> kernel ?G2 ?G1 ?h1" for x
|
|
2497 |
proof -
|
|
2498 |
have xcarr: "x \<in> carrier ?G2"
|
|
2499 |
using that by (auto simp: kernel_def)
|
|
2500 |
interpret G2: comm_group "?G2"
|
|
2501 |
by (rule abelian_relative_homology_group)
|
|
2502 |
let ?b = "hom_boundary p X T x"
|
|
2503 |
have bcarr: "?b \<in> carrier(homology_group(p - 1) (subtopology X T))"
|
|
2504 |
by (simp add: hom_boundary_carrier)
|
|
2505 |
have "hom_boundary p X S (hom_induced p X T X S id x)
|
|
2506 |
= hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id
|
|
2507 |
(hom_boundary p X T x)"
|
|
2508 |
using naturality_hom_induced [of X X id T S p] by (simp add: assms o_def) meson
|
|
2509 |
with bcarr have "hom_boundary p X T x \<in> hom_boundary p (subtopology X S) T ` carrier ?G3"
|
|
2510 |
using homology_exactness_axiom_2 [of p "subtopology X S" T] x
|
|
2511 |
apply (simp add: kernel_def set_eq_iff subtopology_subtopology)
|
|
2512 |
by (metis group_relative_homology_group hom_boundary_hom hom_one set_eq_iff)
|
|
2513 |
then obtain u where ucarr: "u \<in> carrier ?G3"
|
|
2514 |
and ueq: "hom_boundary p X T x = hom_boundary p (subtopology X S) T u"
|
|
2515 |
by (auto simp: kernel_def set_eq_iff subtopology_subtopology hom_boundary_carrier)
|
|
2516 |
define y where "y = x \<otimes>\<^bsub>?G2\<^esub> inv\<^bsub>?G2\<^esub> ?h2 u"
|
|
2517 |
have ycarr: "y \<in> carrier ?G2"
|
|
2518 |
using x by (simp add: y_def kernel_def hom_induced_carrier)
|
|
2519 |
interpret hb: group_hom ?G2 "homology_group (p-1) (subtopology X T)" "hom_boundary p X T"
|
|
2520 |
by (simp add: group_hom_axioms_def group_hom_def hom_boundary_hom)
|
|
2521 |
have yyy: "hom_boundary p X T y = \<one>\<^bsub>homology_group (p - 1) (subtopology X T)\<^esub>"
|
|
2522 |
apply (simp add: y_def bcarr xcarr hom_induced_carrier hom_boundary_carrier hb.inv_solve_right')
|
|
2523 |
using naturality_hom_induced [of concl: p X T "subtopology X S" T id]
|
|
2524 |
apply (simp add: o_def fun_eq_iff subtopology_subtopology)
|
|
2525 |
by (metis hom_boundary_carrier hom_induced_id ueq)
|
|
2526 |
then have "y \<in> hom_induced p X {} X T id ` carrier (homology_group p X)"
|
|
2527 |
using homology_exactness_axiom_1 [of p X T] x ycarr by (auto simp: kernel_def)
|
|
2528 |
then obtain z where zcarr: "z \<in> carrier (homology_group p X)"
|
|
2529 |
and zeq: "hom_induced p X {} X T id z = y"
|
|
2530 |
by auto
|
|
2531 |
interpret gh1: group_hom ?G2 ?G1 ?h1
|
|
2532 |
by (meson group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced)
|
|
2533 |
|
|
2534 |
have "hom_induced p X {} X S id z = (hom_induced p X T X S id \<circ> hom_induced p X {} X T id) z"
|
|
2535 |
by (simp add: assms flip: hom_induced_compose)
|
|
2536 |
also have "\<dots> = \<one>\<^bsub>relative_homology_group p X S\<^esub>"
|
|
2537 |
using x 1 by (simp add: kernel_def zeq y_def)
|
|
2538 |
finally have "hom_induced p X {} X S id z = \<one>\<^bsub>relative_homology_group p X S\<^esub>" .
|
|
2539 |
then have "z \<in> hom_induced p (subtopology X S) {} X {} id `
|
|
2540 |
carrier (homology_group p (subtopology X S))"
|
|
2541 |
using homology_exactness_axiom_3 [of p X S] zcarr by (auto simp: kernel_def)
|
|
2542 |
then obtain w where wcarr: "w \<in> carrier (homology_group p (subtopology X S))"
|
|
2543 |
and weq: "hom_induced p (subtopology X S) {} X {} id w = z"
|
|
2544 |
by blast
|
|
2545 |
let ?u = "hom_induced p (subtopology X S) {} (subtopology X S) T id w \<otimes>\<^bsub>?G3\<^esub> u"
|
|
2546 |
show ?thesis
|
|
2547 |
proof
|
|
2548 |
have *: "x = z \<otimes>\<^bsub>?G2\<^esub> u"
|
|
2549 |
if "z = x \<otimes>\<^bsub>?G2\<^esub> inv\<^bsub>?G2\<^esub> u" "z \<in> carrier ?G2" "u \<in> carrier ?G2" for z u
|
|
2550 |
using that by (simp add: group.inv_solve_right xcarr)
|
|
2551 |
have eq: "?h2 \<circ> hom_induced p (subtopology X S) {} (subtopology X S) T id
|
|
2552 |
= hom_induced p X {} X T id \<circ> hom_induced p (subtopology X S) {} X {} id"
|
|
2553 |
by (simp flip: hom_induced_compose)
|
|
2554 |
show "x = hom_induced p (subtopology X S) T X T id ?u"
|
|
2555 |
apply (simp add: hom_mult [OF hom_induced_hom] hom_induced_carrier ucarr)
|
|
2556 |
apply (rule *)
|
|
2557 |
using eq apply (simp_all add: fun_eq_iff hom_induced_carrier flip: y_def zeq weq)
|
|
2558 |
done
|
|
2559 |
show "?u \<in> carrier (relative_homology_group p (subtopology X S) T)"
|
|
2560 |
by (simp add: abelian_relative_homology_group comm_groupE(1) hom_induced_carrier ucarr)
|
|
2561 |
qed
|
|
2562 |
qed
|
|
2563 |
ultimately show ?thesis
|
|
2564 |
by (auto simp: group_hom_axioms_def group_hom_def hom_induced_hom)
|
|
2565 |
qed
|
|
2566 |
|
|
2567 |
end
|
|
2568 |
|
|
2569 |
|