theory Brouwer_Degree imports Homology_Groups "HOL-Algebra.Multiplicative_Group"
begin
subsection\<open>Reduced Homology\<close>
definition reduced_homology_group :: "int \ 'a topology \ 'a chain set monoid" where"reduced_homology_group p X \
subgroup_generated (homology_group p X)
(kernel (homology_group p X) (homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())))"
lemma one_reduced_homology_group: "\\<^bsub>reduced_homology_group p X\<^esub> = \\<^bsub>homology_group p X\<^esub>" by (simp add: reduced_homology_group_def)
lemma group_reduced_homology_group [simp]: "group (reduced_homology_group p X)" by (simp add: reduced_homology_group_def group.group_subgroup_generated)
lemma carrier_reduced_homology_group: "carrier (reduced_homology_group p X) =
kernel (homology_group p X) (homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()))"
(is"_ = kernel ?G ?H ?h") proof - interpret subgroup "kernel ?G ?H ?h" ?G by (simp add: hom_induced_empty_hom group_hom_axioms_def group_hom_def group_hom.subgroup_kernel) show ?thesis unfolding reduced_homology_group_def using carrier_subgroup_generated_subgroup by blast qed
lemma carrier_reduced_homology_group_subset: "carrier (reduced_homology_group p X) \ carrier (homology_group p X)" by (simp add: group.carrier_subgroup_generated_subset reduced_homology_group_def)
lemma un_reduced_homology_group: assumes"p \ 0" shows"reduced_homology_group p X = homology_group p X" proof - have"(kernel (homology_group p X) (homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())))
= carrier (homology_group p X)" proof (rule group_hom.kernel_to_trivial_group) show"group_hom (homology_group p X) (homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()))" by (auto simp: hom_induced_empty_hom group_hom_def group_hom_axioms_def) show"trivial_group (homology_group p (discrete_topology {()}))" by (simp add: homology_dimension_axiom [OF _ assms]) qed thenshow ?thesis by (simp add: reduced_homology_group_def group.subgroup_generated_group_carrier) qed
lemma trivial_reduced_homology_group: "p < 0 \ trivial_group(reduced_homology_group p X)" by (simp add: trivial_homology_group un_reduced_homology_group)
lemma hom_induced_reduced_hom: "(hom_induced p X {} Y {} f) \ hom (reduced_homology_group p X) (reduced_homology_group p Y)" proof (cases "continuous_map X Y f") case True have eq: "continuous_map X Y f \<Longrightarrow> hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())
= (hom_induced p Y {} (discrete_topology {()}) {} (\<lambda>x. ()) \<circ> hom_induced p X {} Y {} f)" by (simp flip: hom_induced_compose_empty) interpret subgroup "kernel (homology_group p X)
(homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()))" "homology_group p X" by (meson group_hom.subgroup_kernel group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced) have sb: "hom_induced p X {} Y {} f ` carrier (homology_group p X) \ carrier (homology_group p Y)" using hom_induced_carrier by blast show ?thesis using True unfolding reduced_homology_group_def apply (simp add: hom_into_subgroup_eq group_hom.subgroup_kernel hom_induced_empty_hom group.hom_from_subgroup_generated group_hom_def group_hom_axioms_def) unfolding kernel_def using eq sb by auto next case False thenhave"hom_induced p X {} Y {} f = (\c. one(reduced_homology_group p Y))" by (force simp: hom_induced_default reduced_homology_group_def) thenshow ?thesis by (simp add: trivial_hom) qed
lemma hom_induced_reduced: "c \ carrier(reduced_homology_group p X) \<Longrightarrow> hom_induced p X {} Y {} f c \<in> carrier(reduced_homology_group p Y)" by (meson hom_in_carrier hom_induced_reduced_hom)
lemma hom_boundary_reduced_hom: "hom_boundary p X S \<in> hom (relative_homology_group p X S) (reduced_homology_group (p-1) (subtopology X S))" proof - have *: "continuous_map X (discrete_topology {()}) (\x. ())" "(\x. ()) \ S \ {()}" by auto interpret group_hom "relative_homology_group p (discrete_topology {()}) {()}" "homology_group (p-1) (discrete_topology {()})" "hom_boundary p (discrete_topology {()}) {()}" apply (clarsimp simp: group_hom_def group_hom_axioms_def) by (metis UNIV_unit hom_boundary_hom subtopology_UNIV) have"hom_boundary p X S `
carrier (relative_homology_group p X S) \<subseteq> kernel (homology_group (p - 1) (subtopology X S))
(homology_group (p - 1) (discrete_topology {()}))
(hom_induced (p - 1) (subtopology X S) {}
(discrete_topology {()}) {} (\<lambda>x. ()))" proof (clarsimp simp add: kernel_def hom_boundary_carrier) fix c assume c: "c \ carrier (relative_homology_group p X S)" have triv: "trivial_group (relative_homology_group p (discrete_topology {()}) {()})" by (metis topspace_discrete_topology trivial_relative_homology_group_topspace) have"hom_boundary p (discrete_topology {()}) {()}
(hom_induced p X S (discrete_topology {()}) {()} (\<lambda>x. ()) c)
= \<one>\<^bsub>homology_group (p - 1) (discrete_topology {()})\<^esub>" by (metis hom_induced_carrier local.hom_one singletonD triv trivial_group_def) thenshow"hom_induced (p - 1) (subtopology X S) {} (discrete_topology {()}) {} (\x. ()) (hom_boundary p X S c) = \<one>\<^bsub>homology_group (p - 1) (discrete_topology {()})\<^esub>" using naturality_hom_induced [OF *, of p, symmetric] by (simp add: o_def fun_eq_iff) qed thenshow ?thesis by (simp add: reduced_homology_group_def hom_boundary_hom hom_into_subgroup) qed
lemma homotopy_equivalence_reduced_homology_group_isomorphisms: assumes contf: "continuous_map X Y f"and contg: "continuous_map Y X g" and gf: "homotopic_with (\h. True) X X (g \ f) id" and fg: "homotopic_with (\k. True) Y Y (f \ g) id" shows"group_isomorphisms (reduced_homology_group p X) (reduced_homology_group p Y)
(hom_induced p X {} Y {} f) (hom_induced p Y {} X {} g)" proof (simp add: hom_induced_reduced_hom group_isomorphisms_def, intro conjI ballI) fix a assume"a \ carrier (reduced_homology_group p X)" thenhave"(hom_induced p Y {} X {} g \ hom_induced p X {} Y {} f) a = a" apply (simp add: contf contg flip: hom_induced_compose) using carrier_reduced_homology_group_subset gf hom_induced_id homology_homotopy_empty by fastforce thenshow"hom_induced p Y {} X {} g (hom_induced p X {} Y {} f a) = a" by simp next fix b assume"b \ carrier (reduced_homology_group p Y)" thenhave"(hom_induced p X {} Y {} f \ hom_induced p Y {} X {} g) b = b" apply (simp add: contf contg flip: hom_induced_compose) using carrier_reduced_homology_group_subset fg hom_induced_id homology_homotopy_empty by fastforce thenshow"hom_induced p X {} Y {} f (hom_induced p Y {} X {} g b) = b" by (simp add: carrier_reduced_homology_group) qed
lemma homotopy_equivalence_reduced_homology_group_isomorphism: assumes"continuous_map X Y f""continuous_map Y X g" and"homotopic_with (\h. True) X X (g \ f) id" "homotopic_with (\k. True) Y Y (f \ g) id" shows"(hom_induced p X {} Y {} f) \<in> iso (reduced_homology_group p X) (reduced_homology_group p Y)" proof (rule group_isomorphisms_imp_iso) show"group_isomorphisms (reduced_homology_group p X) (reduced_homology_group p Y)
(hom_induced p X {} Y {} f) (hom_induced p Y {} X {} g)" by (simp add: assms homotopy_equivalence_reduced_homology_group_isomorphisms) qed
lemma homotopy_equivalent_space_imp_isomorphic_reduced_homology_groups: "X homotopy_equivalent_space Y \<Longrightarrow> reduced_homology_group p X \<cong> reduced_homology_group p Y" unfolding homotopy_equivalent_space_def using homotopy_equivalence_reduced_homology_group_isomorphism is_isoI by blast
lemma homeomorphic_space_imp_isomorphic_reduced_homology_groups: "X homeomorphic_space Y \ reduced_homology_group p X \ reduced_homology_group p Y" by (simp add: homeomorphic_imp_homotopy_equivalent_space homotopy_equivalent_space_imp_isomorphic_reduced_homology_groups)
lemma trivial_reduced_homology_group_empty: "topspace X = {} \ trivial_group(reduced_homology_group p X)" by (metis carrier_reduced_homology_group_subset group.trivial_group_alt group_reduced_homology_group trivial_group_def trivial_homology_group_empty)
lemma homology_dimension_reduced: assumes"topspace X = {a}" shows"trivial_group (reduced_homology_group p X)" proof - have iso: "(hom_induced p X {} (discrete_topology {()}) {} (\x. ())) \<in> iso (homology_group p X) (homology_group p (discrete_topology {()}))" apply (rule homeomorphic_map_homology_iso) apply (force simp: homeomorphic_map_maps homeomorphic_maps_def assms) done show ?thesis unfolding reduced_homology_group_def by (rule group.trivial_group_subgroup_generated) (use iso in\<open>auto simp: iso_kernel_image\<close>) qed
lemma trivial_reduced_homology_group_contractible_space: "contractible_space X \ trivial_group (reduced_homology_group p X)" apply (simp add: contractible_eq_homotopy_equivalent_singleton_subtopology) apply (auto simp: trivial_reduced_homology_group_empty) using isomorphic_group_triviality by (metis (full_types) group_reduced_homology_group homology_dimension_reduced homotopy_equivalent_space_imp_isomorphic_reduced_homology_groups path_connectedin_def path_connectedin_singleton topspace_subtopology_subset)
lemma image_reduced_homology_group: assumes"topspace X \ S \ {}" shows"hom_induced p X {} X S id ` carrier (reduced_homology_group p X)
= hom_induced p X {} X S id ` carrier (homology_group p X)"
(is"?h ` carrier ?G = ?h ` carrier ?H") proof - obtain a where a: "a \ topspace X" and "a \ S" using assms by blast have [simp]: "A \ {x \ A. P x} = {x \ A. P x}" for A P by blast interpret comm_group "homology_group p X" by (rule abelian_relative_homology_group) have *: "\x'. ?h y = ?h x' \
x' \ carrier ?H \
hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()) x'
= \<one>\<^bsub>homology_group p (discrete_topology {()})\<^esub>" if"y \ carrier ?H" for y proof - let ?f = "hom_induced p (discrete_topology {()}) {} X {} (\x. a)" let ?g = "hom_induced p X {} (discrete_topology {()}) {} (\x. ())" have bcarr: "?f (?g y) \ carrier ?H" by (simp add: hom_induced_carrier) interpret gh1:
group_hom "relative_homology_group p X S""relative_homology_group p (discrete_topology {()}) {()}" "hom_induced p X S (discrete_topology {()}) {()} (\x. ())" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) interpret gh2:
group_hom "relative_homology_group p (discrete_topology {()}) {()}""relative_homology_group p X S" "hom_induced p (discrete_topology {()}) {()} X S (\x. a)" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) interpret gh3:
group_hom "homology_group p X""relative_homology_group p X S""?h" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) interpret gh4:
group_hom "homology_group p X""homology_group p (discrete_topology {()})" "?g" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) interpret gh5:
group_hom "homology_group p (discrete_topology {()})""homology_group p X" "?f" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) interpret gh6:
group_hom "homology_group p (discrete_topology {()})""relative_homology_group p (discrete_topology {()}) {()}" "hom_induced p (discrete_topology {()}) {} (discrete_topology {()}) {()} id" by (meson group_hom_axioms_def group_hom_def hom_induced_hom group_relative_homology_group) show ?thesis proof (intro exI conjI) have"(?h \ ?f \ ?g) y
= (hom_induced p (discrete_topology {()}) {()} X S (\<lambda>x. a) \<circ>
hom_induced p (discrete_topology {()}) {} (discrete_topology {()}) {()} id \<circ> ?g) y" by (simp add: a \<open>a \<in> S\<close> flip: hom_induced_compose) alsohave"\ = \\<^bsub>relative_homology_group p X S\<^esub>" using trivial_relative_homology_group_topspace [of p "discrete_topology {()}"] apply simp by (metis (full_types) empty_iff gh1.H.one_closed gh1.H.trivial_group gh2.hom_one hom_induced_carrier insert_iff) finallyhave"?h (?f (?g y)) = \\<^bsub>relative_homology_group p X S\<^esub>" by simp thenshow"?h y = ?h (y \\<^bsub>?H\<^esub> inv\<^bsub>?H\<^esub> ?f (?g y))" by (simp add: that hom_induced_carrier) show"(y \\<^bsub>?H\<^esub> inv\<^bsub>?H\<^esub> ?f (?g y)) \ carrier (homology_group p X)" by (simp add: hom_induced_carrier that) have *: "(?g \ hom_induced p X {} X {} (\x. a)) y = hom_induced p X {} (discrete_topology {()}) {} (\a. ()) y" by (simp add: a \<open>a \<in> S\<close> flip: hom_induced_compose) have"?g (y \\<^bsub>?H\<^esub> inv\<^bsub>?H\<^esub> (?f \ ?g) y)
= \<one>\<^bsub>homology_group p (discrete_topology {()})\<^esub>" by (simp add: a \<open>a \<in> S\<close> that hom_induced_carrier flip: hom_induced_compose * [unfolded o_def]) thenshow"?g (y \\<^bsub>?H\<^esub> inv\<^bsub>?H\<^esub> ?f (?g y))
= \<one>\<^bsub>homology_group p (discrete_topology {()})\<^esub>" by simp qed qed show ?thesis apply (auto simp: reduced_homology_group_def carrier_subgroup_generated kernel_def image_iff) apply (metis (no_types, lifting) generate_in_carrier mem_Collect_eq subsetI) apply (force simp: dest: * intro: generate.incl) done qed
lemma homology_exactness_reduced_1: assumes"topspace X \ S \ {}" shows"exact_seq([reduced_homology_group(p - 1) (subtopology X S),
relative_homology_group p X S,
reduced_homology_group p X],
[hom_boundary p X S, hom_induced p X {} X S id])"
(is"exact_seq ([?G1,?G2,?G3], [?h1,?h2])") proof - have *: "?h2 ` carrier (homology_group p X)
= kernel ?G2 (homology_group (p - 1) (subtopology X S)) ?h1" using homology_exactness_axiom_1 [of p X S] by simp have gh: "group_hom ?G3 ?G2 ?h2" by (simp add: reduced_homology_group_def group_hom_def group_hom_axioms_def
group.group_subgroup_generated group.hom_from_subgroup_generated hom_induced_hom) show ?thesis apply (simp add: hom_boundary_reduced_hom gh * image_reduced_homology_group [OF assms]) apply (simp add: kernel_def one_reduced_homology_group) done qed
lemma homology_exactness_reduced_2: "exact_seq([reduced_homology_group(p - 1) X,
reduced_homology_group(p - 1) (subtopology X S),
relative_homology_group p X S],
[hom_induced (p - 1) (subtopology X S) {} X {} id, hom_boundary p X S])"
(is"exact_seq ([?G1,?G2,?G3], [?h1,?h2])") using homology_exactness_axiom_2 [of p X S] apply (simp add: group_hom_axioms_def group_hom_def hom_boundary_reduced_hom hom_induced_reduced_hom) apply (simp add: reduced_homology_group_def group_hom.subgroup_kernel group_hom_axioms_def group_hom_def hom_induced_hom) using hom_boundary_reduced_hom [of p X S] apply (auto simp: image_def set_eq_iff) by (metis carrier_reduced_homology_group hom_in_carrier set_eq_iff)
lemma homology_exactness_reduced_3: "exact_seq([relative_homology_group p X S,
reduced_homology_group p X,
reduced_homology_group p (subtopology X S)],
[hom_induced p X {} X S id, hom_induced p (subtopology X S) {} X {} id])"
(is"exact_seq ([?G1,?G2,?G3], [?h1,?h2])") proof - have"kernel ?G2 ?G1 ?h1 =
?h2 ` carrier ?G3" proof - obtain U where U: "(hom_induced p (subtopology X S) {} X {} id) ` carrier ?G3 \ U" "(hom_induced p (subtopology X S) {} X {} id) ` carrier ?G3 \<subseteq> (hom_induced p (subtopology X S) {} X {} id) ` carrier (homology_group p (subtopology X S))" "U \ kernel (homology_group p X) ?G1 (hom_induced p X {} X S id)
= kernel ?G2 ?G1 (hom_induced p X {} X S id)" "U \ (hom_induced p (subtopology X S) {} X {} id) ` carrier (homology_group p (subtopology X S)) \<subseteq> (hom_induced p (subtopology X S) {} X {} id) ` carrier ?G3" proof show"?h2 ` carrier ?G3 \ carrier ?G2" by (simp add: hom_induced_reduced image_subset_iff) show"?h2 ` carrier ?G3 \ ?h2 ` carrier (homology_group p (subtopology X S))" by (meson carrier_reduced_homology_group_subset image_mono) have"subgroup (kernel (homology_group p X) (homology_group p (discrete_topology {()}))
(hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())))
(homology_group p X)" by (simp add: group.normal_invE(1) group_hom.normal_kernel group_hom_axioms_def group_hom_def hom_induced_empty_hom) thenshow"carrier ?G2 \ kernel (homology_group p X) ?G1 ?h1 = kernel ?G2 ?G1 ?h1" unfolding carrier_reduced_homology_group by (auto simp: reduced_homology_group_def) show"carrier ?G2 \ ?h2 ` carrier (homology_group p (subtopology X S)) \<subseteq> ?h2 ` carrier ?G3" by (force simp: carrier_reduced_homology_group kernel_def hom_induced_compose') qed with homology_exactness_axiom_3 [of p X S] show ?thesis by (fastforce simp add:) qed thenshow ?thesis apply (simp add: group_hom_axioms_def group_hom_def hom_boundary_reduced_hom hom_induced_reduced_hom) apply (simp add: group.hom_from_subgroup_generated hom_induced_hom reduced_homology_group_def) done qed
subsection\<open>More homology properties of deformations, retracts, contractible spaces\<close>
lemma iso_relative_homology_of_contractible: "\contractible_space X; topspace X \ S \ {}\ \<Longrightarrow> hom_boundary p X S \<in> iso (relative_homology_group p X S) (reduced_homology_group(p - 1) (subtopology X S))" using very_short_exact_sequence
[of "reduced_homology_group (p - 1) X" "reduced_homology_group (p - 1) (subtopology X S)" "relative_homology_group p X S" "reduced_homology_group p X" "hom_induced (p - 1) (subtopology X S) {} X {} id" "hom_boundary p X S" "hom_induced p X {} X S id"] by (meson exact_seq_cons_iff homology_exactness_reduced_1 homology_exactness_reduced_2 trivial_reduced_homology_group_contractible_space)
lemma isomorphic_group_relative_homology_of_contractible: "\contractible_space X; topspace X \ S \ {}\ \<Longrightarrow> relative_homology_group p X S \<cong>
reduced_homology_group(p - 1) (subtopology X S)" by (meson iso_relative_homology_of_contractible is_isoI)
lemma isomorphic_group_reduced_homology_of_contractible: "\contractible_space X; topspace X \ S \ {}\ \<Longrightarrow> reduced_homology_group p (subtopology X S) \<cong> relative_homology_group(p + 1) X S" by (metis add.commute add_diff_cancel_left' group.iso_sym group_relative_homology_group isomorphic_group_relative_homology_of_contractible)
lemma iso_reduced_homology_by_contractible: "\contractible_space(subtopology X S); topspace X \ S \ {}\ \<Longrightarrow> (hom_induced p X {} X S id) \<in> iso (reduced_homology_group p X) (relative_homology_group p X S)" using very_short_exact_sequence
[of "reduced_homology_group (p - 1) (subtopology X S)" "relative_homology_group p X S" "reduced_homology_group p X" "reduced_homology_group p (subtopology X S)" "hom_boundary p X S" "hom_induced p X {} X S id" "hom_induced p (subtopology X S) {} X {} id"] by (meson exact_seq_cons_iff homology_exactness_reduced_1 homology_exactness_reduced_3 trivial_reduced_homology_group_contractible_space)
lemma isomorphic_reduced_homology_by_contractible: "\contractible_space(subtopology X S); topspace X \ S \ {}\ \<Longrightarrow> reduced_homology_group p X \<cong> relative_homology_group p X S" using is_isoI iso_reduced_homology_by_contractible by blast
lemma isomorphic_relative_homology_by_contractible: "\contractible_space(subtopology X S); topspace X \ S \ {}\ \<Longrightarrow> relative_homology_group p X S \<cong> reduced_homology_group p X" using group.iso_sym group_reduced_homology_group isomorphic_reduced_homology_by_contractible by blast
lemma isomorphic_reduced_homology_by_singleton: "a \ topspace X \ reduced_homology_group p X \ relative_homology_group p X ({a})" by (simp add: contractible_space_subtopology_singleton isomorphic_reduced_homology_by_contractible)
lemma isomorphic_relative_homology_by_singleton: "a \ topspace X \ relative_homology_group p X ({a}) \ reduced_homology_group p X" by (simp add: group.iso_sym isomorphic_reduced_homology_by_singleton)
lemma reduced_homology_group_pair: assumes"t1_space X"and a: "a \ topspace X" and b: "b \ topspace X" and "a \ b" shows"reduced_homology_group p (subtopology X {a,b}) \ homology_group p (subtopology X {a})"
(is"?lhs \ ?rhs") proof - have"?lhs \ relative_homology_group p (subtopology X {a,b}) {b}" by (simp add: b isomorphic_reduced_homology_by_singleton topspace_subtopology) alsohave"\ \ ?rhs" proof - have sub: "subtopology X {a, b} closure_of {b} \ subtopology X {a, b} interior_of {b}" by (simp add: assms t1_space_subtopology closure_of_singleton subtopology_eq_discrete_topology_finite discrete_topology_closure_of) show ?thesis using homology_excision_axiom [OF sub, of "{a,b}" p] by (simp add: assms(4) group.iso_sym is_isoI subtopology_subtopology) qed finallyshow ?thesis . qed
lemma deformation_retraction_relative_homology_group_isomorphisms: "\retraction_maps X Y r s; r \ U \ V; s \ V \ U; homotopic_with (\h. h ` U \ U) X X (s \ r) id\ \<Longrightarrow> group_isomorphisms (relative_homology_group p X U) (relative_homology_group p Y V)
(hom_induced p X U Y V r) (hom_induced p Y V X U s)" apply (simp add: retraction_maps_def) apply (rule homotopy_equivalence_relative_homology_group_isomorphisms) apply (auto simp: image_subset_iff_funcset Pi_iff continuous_map_compose homotopic_with_equal) done
lemma deformation_retract_relative_homology_group_isomorphisms: "\retraction_maps X Y r id; V \ U; r \ U \ V; homotopic_with (\h. h ` U \ U) X X r id\ \<Longrightarrow> group_isomorphisms (relative_homology_group p X U) (relative_homology_group p Y V)
(hom_induced p X U Y V r) (hom_induced p Y V X U id)" by (simp add: deformation_retraction_relative_homology_group_isomorphisms
in_mono)
lemma deformation_retract_relative_homology_group_isomorphism: "\retraction_maps X Y r id; V \ U; r \ U \ V; homotopic_with (\h. h ` U \ U) X X r id\ \<Longrightarrow> (hom_induced p X U Y V r) \<in> iso (relative_homology_group p X U) (relative_homology_group p Y V)" by (metis deformation_retract_relative_homology_group_isomorphisms group_isomorphisms_imp_iso)
lemma deformation_retract_relative_homology_group_isomorphism_id: "\retraction_maps X Y r id; V \ U; r \ U \ V; homotopic_with (\h. h ` U \ U) X X r id\ \<Longrightarrow> (hom_induced p Y V X U id) \<in> iso (relative_homology_group p Y V) (relative_homology_group p X U)" by (metis deformation_retract_relative_homology_group_isomorphisms group_isomorphisms_imp_iso group_isomorphisms_sym)
lemma deformation_retraction_imp_isomorphic_relative_homology_groups: "\retraction_maps X Y r s; r \ U \ V; s ` V \ U; homotopic_with (\h. h ` U \ U) X X (s \ r) id\ \<Longrightarrow> relative_homology_group p X U \<cong> relative_homology_group p Y V" by (blast intro: is_isoI group_isomorphisms_imp_iso deformation_retraction_relative_homology_group_isomorphisms)
lemma deformation_retraction_imp_isomorphic_homology_groups: "\retraction_maps X Y r s; homotopic_with (\h. True) X X (s \ r) id\ \<Longrightarrow> homology_group p X \<cong> homology_group p Y" by (simp add: deformation_retraction_imp_homotopy_equivalent_space homotopy_equivalent_space_imp_isomorphic_homology_groups)
lemma deformation_retract_imp_isomorphic_relative_homology_groups: "\retraction_maps X X' r id; V \ U; r \ U \ V; homotopic_with (\h. h ` U \ U) X X r id\ \<Longrightarrow> relative_homology_group p X U \<cong> relative_homology_group p X' V" by (simp add: deformation_retraction_imp_isomorphic_relative_homology_groups)
lemma deformation_retract_imp_isomorphic_homology_groups: "\retraction_maps X X' r id; homotopic_with (\h. True) X X r id\ \<Longrightarrow> homology_group p X \<cong> homology_group p X'" by (simp add: deformation_retraction_imp_isomorphic_homology_groups)
lemma epi_hom_induced_inclusion: assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" shows"(hom_induced p (subtopology X S) {} X {} id) \<in> epi (homology_group p (subtopology X S)) (homology_group p X)" proof (rule epi_right_invertible) show"hom_induced p (subtopology X S) {} X {} id \<in> hom (homology_group p (subtopology X S)) (homology_group p X)" by (simp add: hom_induced_empty_hom) show"hom_induced p X {} (subtopology X S) {} f \<in> carrier (homology_group p X) \<rightarrow> carrier (homology_group p (subtopology X S))" by (simp add: hom_induced_carrier) fix x assume x: "x \ carrier (homology_group p X)" show"hom_induced p (subtopology X S) {} X {} id (hom_induced p X {} (subtopology X S) {} f x) = x" proof (subst hom_induced_compose') show"continuous_map X (subtopology X S) f" by (meson assms continuous_map_into_subtopology
homotopic_with_imp_continuous_maps) show"hom_induced p X {} X {} (id \ f) x = x" by (metis assms(1) hom_induced_id homology_homotopy_empty id_comp x) qed (use assms in auto) qed
lemma trivial_homomorphism_hom_induced_relativization: assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" shows"trivial_homomorphism (homology_group p X) (relative_homology_group p X S)
(hom_induced p X {} X S id)" proof - have"(hom_induced p (subtopology X S) {} X {} id) \<in> epi (homology_group p (subtopology X S)) (homology_group p X)" by (metis assms epi_hom_induced_inclusion) thenshow ?thesis using homology_exactness_axiom_3 [of p X S] homology_exactness_axiom_1 [of p X S] by (simp add: epi_def group.trivial_homomorphism_image group_hom.trivial_hom_iff) qed
lemma mon_hom_boundary_inclusion: assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" shows"(hom_boundary p X S) \ mon
(relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))" proof - have"(hom_induced p (subtopology X S) {} X {} id) \<in> epi (homology_group p (subtopology X S)) (homology_group p X)" by (metis assms epi_hom_induced_inclusion) thenshow ?thesis using homology_exactness_axiom_3 [of p X S] homology_exactness_axiom_1 [of p X S] apply (simp add: mon_def epi_def hom_boundary_hom) by (metis (no_types, opaque_lifting) group_hom.trivial_hom_iff group_hom.trivial_ker_imp_inj group_hom_axioms_def group_hom_def group_relative_homology_group hom_boundary_hom) qed
lemma short_exact_sequence_hom_induced_relativization: assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" shows"short_exact_sequence (homology_group (p-1) X) (homology_group (p-1) (subtopology X S)) (relative_homology_group p X S)
(hom_induced (p-1) (subtopology X S) {} X {} id) (hom_boundary p X S)" unfolding short_exact_sequence_iff by (intro conjI homology_exactness_axiom_2 epi_hom_induced_inclusion [OF assms] mon_hom_boundary_inclusion [OF assms])
lemma group_isomorphisms_homology_group_prod_deformation: fixes p::int assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" obtains H K where "subgroup H (homology_group p (subtopology X S))" "subgroup K (homology_group p (subtopology X S))" "(\(x, y). x \\<^bsub>homology_group p (subtopology X S)\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) H \<times>\<times>
subgroup_generated (homology_group p (subtopology X S)) K)
(homology_group p (subtopology X S))" "hom_boundary (p + 1) X S \<in> Group.iso (relative_homology_group (p + 1) X S)
(subgroup_generated (homology_group p (subtopology X S)) H)" "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso
(subgroup_generated (homology_group p (subtopology X S)) K)
(homology_group p X)" proof - let ?rhs = "relative_homology_group (p + 1) X S" let ?pXS = "homology_group p (subtopology X S)" let ?pX = "homology_group p X" let ?hb = "hom_boundary (p + 1) X S" let ?hi = "hom_induced p (subtopology X S) {} X {} id" have x: "short_exact_sequence (?pX) ?pXS ?rhs ?hi ?hb" using short_exact_sequence_hom_induced_relativization [OF assms, of "p + 1"] by simp have contf: "continuous_map X (subtopology X S) f" by (metis assms continuous_map_into_subtopology homotopic_with_imp_continuous_maps) obtain H K where HK: "H \ ?pXS" "subgroup K ?pXS" "H \ K \ {one ?pXS}" "set_mult ?pXS H K = carrier ?pXS" and iso: "?hb \ iso ?rhs (subgroup_generated ?pXS H)" "?hi \ iso (subgroup_generated ?pXS K) ?pX" proof (rule splitting_lemma_right [OF x, where g' = "hom_induced p X {} (subtopology X S) {} f"]) show"hom_induced p X {} (subtopology X S) {} f \ hom (homology_group p X) (homology_group p (subtopology X S))" using hom_induced_empty_hom by blast next fix z assume"z \ carrier (homology_group p X)" thenshow"hom_induced p (subtopology X S) {} X {} id (hom_induced p X {} (subtopology X S) {} f z) = z" using assms(1) contf hom_induced_id homology_homotopy_empty by (fastforce simp add: hom_induced_compose') qed blast show ?thesis proof show"subgroup H ?pXS" using HK(1) normal_imp_subgroup by blast thenshow"(\(x, y). x \\<^bsub>?pXS\<^esub> y) \<in> Group.iso (subgroup_generated (?pXS) H \<times>\<times> subgroup_generated (?pXS) K) (?pXS)" by (meson HK abelian_relative_homology_group group_disjoint_sum.iso_group_mul group_disjoint_sum_def group_relative_homology_group) show"subgroup K ?pXS" by (rule HK) show"hom_boundary (p + 1) X S \ Group.iso ?rhs (subgroup_generated (?pXS) H)" using iso int_ops(4) by presburger show"hom_induced p (subtopology X S) {} X {} id \ Group.iso (subgroup_generated (?pXS) K) (?pX)" by (simp add: iso(2)) qed qed
lemma iso_homology_group_prod_deformation: assumes"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" shows"homology_group p (subtopology X S) \<cong> DirProd (homology_group p X) (relative_homology_group(p + 1) X S)"
(is"?G \ DirProd ?H ?R") proof - obtain H K where HK: "(\(x, y). x \\<^bsub>?G\<^esub> y) \<in> Group.iso (subgroup_generated (?G) H \<times>\<times> subgroup_generated (?G) K) (?G)" "hom_boundary (p + 1) X S \ Group.iso (?R) (subgroup_generated (?G) H)" "hom_induced p (subtopology X S) {} X {} id \ Group.iso (subgroup_generated (?G) K) (?H)" by (blast intro: group_isomorphisms_homology_group_prod_deformation [OF assms]) have"?G \ DirProd (subgroup_generated (?G) H) (subgroup_generated (?G) K)" by (meson DirProd_group HK(1) group.group_subgroup_generated group.iso_sym group_relative_homology_group is_isoI) alsohave"\ \ DirProd ?R ?H" by (meson HK group.DirProd_iso_trans group.group_subgroup_generated group.iso_sym group_relative_homology_group is_isoI) alsohave"\ \ DirProd ?H ?R" by (simp add: DirProd_commute_iso) finallyshow ?thesis . qed
lemma iso_homology_contractible_space_subtopology1: assumes"contractible_space X""S \ topspace X" "S \ {}" shows"homology_group 0 (subtopology X S) \ DirProd integer_group (relative_homology_group(1) X S)" proof - obtain f where"homotopic_with (\x. True) X X id f" and "f \ topspace X \ S" using assms contractible_space_alt by fastforce thenhave"homology_group 0 (subtopology X S) \ homology_group 0 X \\ relative_homology_group 1 X S" using iso_homology_group_prod_deformation [of X _ S 0] by auto alsohave"\ \ integer_group \\ relative_homology_group 1 X S" using assms contractible_imp_path_connected_space group.DirProd_iso_trans group_relative_homology_group iso_refl isomorphic_integer_zeroth_homology_group by blast finallyshow ?thesis . qed
lemma iso_homology_contractible_space_subtopology2: "\contractible_space X; S \ topspace X; p \ 0; S \ {}\ \<Longrightarrow> homology_group p (subtopology X S) \<cong> relative_homology_group (p + 1) X S" by (metis (no_types, opaque_lifting) add.commute isomorphic_group_reduced_homology_of_contractible topspace_subtopology topspace_subtopology_subset un_reduced_homology_group)
lemma trivial_relative_homology_group_contractible_spaces: "\contractible_space X; contractible_space(subtopology X S); topspace X \ S \ {}\ \<Longrightarrow> trivial_group(relative_homology_group p X S)" using group_reduced_homology_group group_relative_homology_group isomorphic_group_triviality isomorphic_relative_homology_by_contractible trivial_reduced_homology_group_contractible_space by blast
lemma trivial_relative_homology_group_alt: assumes contf: "continuous_map X (subtopology X S) f"and hom: "homotopic_with (\k. k ` S \ S) X X f id" shows"trivial_group (relative_homology_group p X S)" proof (rule trivial_relative_homology_group_gen [OF contf]) show"homotopic_with (\h. True) (subtopology X S) (subtopology X S) f id" using hom unfolding homotopic_with_def apply (rule ex_forward) apply (auto simp: prod_topology_subtopology continuous_map_in_subtopology continuous_map_from_subtopology image_subset_iff topspace_subtopology) done show"homotopic_with (\k. True) X X f id" using assms by (force simp: homotopic_with_def) qed
lemma iso_hom_induced_relativization_contractible: assumes"contractible_space(subtopology X S)""contractible_space(subtopology X T)""T \ S" "topspace X \ T \ {}" shows"(hom_induced p X T X S id) \ iso (relative_homology_group p X T) (relative_homology_group p X S)" proof (rule very_short_exact_sequence) show"exact_seq
([relative_homology_group(p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T],
[hom_relboundary p X S T, hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])" using homology_exactness_triple_1 [OF \<open>T \<subseteq> S\<close>] homology_exactness_triple_3 [OF \<open>T \<subseteq> S\<close>] by fastforce show"trivial_group (relative_homology_group p (subtopology X S) T)""trivial_group (relative_homology_group(p - 1) (subtopology X S) T)" using assms by (force simp: inf.absorb_iff2 subtopology_subtopology topspace_subtopology intro!: trivial_relative_homology_group_contractible_spaces)+ qed
corollary isomorphic_relative_homology_groups_relativization_contractible: assumes"contractible_space(subtopology X S)""contractible_space(subtopology X T)""T \ S" "topspace X \ T \ {}" shows"relative_homology_group p X T \ relative_homology_group p X S" by (rule is_isoI) (rule iso_hom_induced_relativization_contractible [OF assms])
lemma iso_hom_induced_inclusion_contractible: assumes"contractible_space X""contractible_space(subtopology X S)""T \ S" "topspace X \ S \ {}" shows"(hom_induced p (subtopology X S) T X T id) \<in> iso (relative_homology_group p (subtopology X S) T) (relative_homology_group p X T)" proof (rule very_short_exact_sequence) show"exact_seq
([relative_homology_group p X S, relative_homology_group p X T,
relative_homology_group p (subtopology X S) T, relative_homology_group (p+1) X S],
[hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id, hom_relboundary (p+1) X S T])" using homology_exactness_triple_2 [OF \<open>T \<subseteq> S\<close>] homology_exactness_triple_3 [OF \<open>T \<subseteq> S\<close>] by (metis add_diff_cancel_left' diff_add_cancel exact_seq_cons_iff) show"trivial_group (relative_homology_group (p+1) X S)""trivial_group (relative_homology_group p X S)" using assms by (auto simp: subtopology_subtopology topspace_subtopology intro!: trivial_relative_homology_group_contractible_spaces) qed
corollary isomorphic_relative_homology_groups_inclusion_contractible: assumes"contractible_space X""contractible_space(subtopology X S)""T \ S" "topspace X \ S \ {}" shows"relative_homology_group p (subtopology X S) T \ relative_homology_group p X T" by (rule is_isoI) (rule iso_hom_induced_inclusion_contractible [OF assms])
lemma iso_hom_relboundary_contractible: assumes"contractible_space X""contractible_space(subtopology X T)""T \ S" "topspace X \ T \ {}" shows"hom_relboundary p X S T \<in> iso (relative_homology_group p X S) (relative_homology_group (p - 1) (subtopology X S) T)" proof (rule very_short_exact_sequence) show"exact_seq
([relative_homology_group (p - 1) X T, relative_homology_group (p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T],
[hom_induced (p - 1) (subtopology X S) T X T id, hom_relboundary p X S T, hom_induced p X T X S id])" using homology_exactness_triple_1 [OF \<open>T \<subseteq> S\<close>] homology_exactness_triple_2 [OF \<open>T \<subseteq> S\<close>] by simp show"trivial_group (relative_homology_group p X T)""trivial_group (relative_homology_group (p - 1) X T)" using assms by (auto simp: subtopology_subtopology topspace_subtopology intro!: trivial_relative_homology_group_contractible_spaces) qed
corollary isomorphic_relative_homology_groups_relboundary_contractible: assumes"contractible_space X""contractible_space(subtopology X T)""T \ S" "topspace X \ T \ {}" shows"relative_homology_group p X S \ relative_homology_group (p - 1) (subtopology X S) T" by (rule is_isoI) (rule iso_hom_relboundary_contractible [OF assms])
lemma isomorphic_relative_contractible_space_imp_homology_groups: assumes"contractible_space X""contractible_space Y""S \ topspace X" "T \ topspace Y" and ST: "S = {} \ T = {}" and iso: "\p. relative_homology_group p X S \ relative_homology_group p Y T" shows"homology_group p (subtopology X S) \ homology_group p (subtopology Y T)" proof (cases "T = {}") case True have"homology_group p (subtopology X {}) \ homology_group p (subtopology Y {})" by (simp add: homeomorphic_empty_space_eq homeomorphic_space_imp_isomorphic_homology_groups) thenshow ?thesis using ST True by blast next case False show ?thesis proof (cases "p = 0") case True have"homology_group p (subtopology X S) \ integer_group \\ relative_homology_group 1 X S" using assms True \<open>T \<noteq> {}\<close> by (simp add: iso_homology_contractible_space_subtopology1) alsohave"\ \ integer_group \\ relative_homology_group 1 Y T" by (simp add: assms group.DirProd_iso_trans iso_refl) alsohave"\ \ homology_group p (subtopology Y T)" by (simp add: True \<open>T \<noteq> {}\<close> assms group.iso_sym iso_homology_contractible_space_subtopology1) finallyshow ?thesis . next case False have"homology_group p (subtopology X S) \ relative_homology_group (p+1) X S" using assms False \<open>T \<noteq> {}\<close> by (simp add: iso_homology_contractible_space_subtopology2) alsohave"\ \ relative_homology_group (p+1) Y T" by (simp add: assms) alsohave"\ \ homology_group p (subtopology Y T)" by (simp add: False \<open>T \<noteq> {}\<close> assms group.iso_sym iso_homology_contractible_space_subtopology2) finallyshow ?thesis . qed qed
subsection\<open>Homology groups of spheres\<close>
lemma iso_reduced_homology_group_lower_hemisphere: assumes"k \ n" shows"hom_induced p (nsphere n) {} (nsphere n) {x. x k \ 0} id \<in> iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. x k \<le> 0})" proof (rule iso_reduced_homology_by_contractible) show"contractible_space (subtopology (nsphere n) {x. x k \ 0})" by (simp add: assms contractible_space_lower_hemisphere) have"(\i. if i = k then -1 else 0) \ topspace (nsphere n) \ {x. x k \ 0}" using assms by (simp add: nsphere if_distrib [of "\x. x ^ 2"] cong: if_cong) thenshow"topspace (nsphere n) \ {x. x k \ 0} \ {}" by blast qed
lemma topspace_nsphere_1: assumes"x \ topspace (nsphere n)" shows "(x k)\<^sup>2 \ 1" proof (cases "k \ n") case True have"(\i \ {..n} - {k}. (x i)\<^sup>2) = (\i\n. (x i)\<^sup>2) - (x k)\<^sup>2" using\<open>k \<le> n\<close> by (simp add: sum_diff) thenshow ?thesis using assms apply (simp add: nsphere) by (metis diff_ge_0_iff_ge sum_nonneg zero_le_power2) next case False thenshow ?thesis using assms by (simp add: nsphere) qed
lemma topspace_nsphere_1_eq_0: fixes x :: "nat \ real" assumes x: "x \ topspace (nsphere n)" and xk: "(x k)\<^sup>2 = 1" and "i \ k" shows"x i = 0" proof (cases "i \ n") case True have"k \ n" using x by (simp add: nsphere) (metis not_less xk zero_neq_one zero_power2) have"(\i \ {..n} - {k}. (x i)\<^sup>2) = (\i\n. (x i)\<^sup>2) - (x k)\<^sup>2" using\<open>k \<le> n\<close> by (simp add: sum_diff) alsohave"\ = 0" using assms by (simp add: nsphere) finallyhave"\i\{..n} - {k}. (x i)\<^sup>2 = 0" by (simp add: sum_nonneg_eq_0_iff) thenshow ?thesis using True \<open>i \<noteq> k\<close> by auto next case False with x show ?thesis by (simp add: nsphere) qed
proposition iso_relative_homology_group_upper_hemisphere: "(hom_induced p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0} (nsphere n) {x. x k \ 0} id) \<in> iso (relative_homology_group p (subtopology (nsphere n) {x. x k \<ge> 0}) {x. x k = 0})
(relative_homology_group p (nsphere n) {x. x k \<le> 0})" (is "?h \<in> iso ?G ?H") proof - have"topspace (nsphere n) \ {x. x k < - 1 / 2} \ {x \ topspace (nsphere n). x k \ {y. y \ - 1 / 2}}" by force moreoverhave"closedin (nsphere n) {x \ topspace (nsphere n). x k \ {y. y \ - 1 / 2}}" apply (rule closedin_continuous_map_preimage [OF continuous_map_nsphere_projection]) using closed_Collect_le [of id "\x::real. -1/2"] apply simp done ultimatelyhave"nsphere n closure_of {x. x k < -1/2} \ {x \ topspace (nsphere n). x k \ {y. y \ -1/2}}" by (metis (no_types, lifting) closure_of_eq closure_of_mono closure_of_restrict) alsohave"\ \ {x \ topspace (nsphere n). x k \ {y. y < 0}}" by force alsohave"\ \ nsphere n interior_of {x. x k \ 0}" proof (rule interior_of_maximal) show"{x \ topspace (nsphere n). x k \ {y. y < 0}} \ {x. x k \ 0}" by force show"openin (nsphere n) {x \ topspace (nsphere n). x k \ {y. y < 0}}" apply (rule openin_continuous_map_preimage [OF continuous_map_nsphere_projection]) using open_Collect_less [of id "\x::real. 0"] apply simp done qed finallyhave nn: "nsphere n closure_of {x. x k < -1/2} \ nsphere n interior_of {x. x k \ 0}" . have [simp]: "{x::nat\real. x k \ 0} - {x. x k < - (1/2)} = {x. -1/2 \ x k \ x k \ 0}" "UNIV - {x::nat\real. x k < a} = {x. a \ x k}" for a by auto let ?T01 = "top_of_set {0..1::real}" let ?X12 = "subtopology (nsphere n) {x. -1/2 \ x k}" have 1: "hom_induced p ?X12 {x. -1/2 \ x k \ x k \ 0} (nsphere n) {x. x k \ 0} id \<in> iso (relative_homology_group p ?X12 {x. -1/2 \<le> x k \<and> x k \<le> 0})
?H" using homology_excision_axiom [OF nn subset_UNIV, of p] by simp
define h where"h \ \(T,x). let y = max (x k) (-T) in
(\<lambda>i. if i = k then y else sqrt(1 - y ^ 2) / sqrt(1 - x k ^ 2) * x i)" have h: "h(T,x) = x"if"0 \ T" "T \ 1" "(\i\n. (x i)\<^sup>2) = 1" and 0: "\i>n. x i = 0" "-T \ x k" for T x using that by (force simp: nsphere h_def Let_def max_def intro!: topspace_nsphere_1_eq_0) have"continuous_map (prod_topology ?T01 ?X12) euclideanreal (\x. h x i)" for i proof - show ?thesis proof (rule continuous_map_eq) show"continuous_map (prod_topology ?T01 ?X12)
euclideanreal (\<lambda>(T, x). if 0 \<le> x k then x i else h (T, x) i)" unfolding case_prod_unfold proof (rule continuous_map_cases_le) show"continuous_map (prod_topology ?T01 ?X12) euclideanreal (\x. snd x k)" apply (subst continuous_map_of_snd [unfolded o_def]) by (simp add: continuous_map_from_subtopology continuous_map_nsphere_projection) next show"continuous_map (subtopology (prod_topology ?T01 ?X12) {p \ topspace (prod_topology ?T01 ?X12). 0 \ snd p k})
euclideanreal (\<lambda>x. snd x i)" apply (rule continuous_map_from_subtopology) apply (subst continuous_map_of_snd [unfolded o_def]) by (simp add: continuous_map_from_subtopology continuous_map_nsphere_projection) next note fst = continuous_map_into_fulltopology [OF continuous_map_subtopology_fst] have snd: "continuous_map (subtopology (prod_topology ?T01 (subtopology (nsphere n) T)) S) euclideanreal (\x. snd x k)" for k S T apply (simp add: nsphere) apply (rule continuous_map_from_subtopology) apply (subst continuous_map_of_snd [unfolded o_def]) using continuous_map_from_subtopology continuous_map_nsphere_projection nsphere by fastforce show"continuous_map (subtopology (prod_topology ?T01 ?X12) {p \ topspace (prod_topology ?T01 ?X12). snd p k \ 0})
euclideanreal (\<lambda>x. h (fst x, snd x) i)" apply (simp add: h_def case_prod_unfold Let_def) apply (intro conjI impI fst snd continuous_intros) apply (auto simp: nsphere power2_eq_1_iff) done qed (auto simp: nsphere h) qed (auto simp: nsphere h) qed moreover have"h ` ({0..1} \ (topspace (nsphere n) \ {x. - (1/2) \ x k})) \<subseteq> {x. (\<Sum>i\<le>n. (x i)\<^sup>2) = 1 \<and> (\<forall>i>n. x i = 0)}" proof - have"(\i\n. (h (T,x) i)\<^sup>2) = 1" if x: "x \ topspace (nsphere n)" and xk: "- (1/2) \ x k" and T: "0 \ T" "T \ 1" for T x proof (cases "-T \ x k ") case True thenshow ?thesis using that by (auto simp: nsphere h) next case False with x \<open>0 \<le> T\<close> have "k \<le> n" apply (simp add: nsphere) by (metis neg_le_0_iff_le not_le) have"1 - (x k)\<^sup>2 \ 0" using topspace_nsphere_1 x by auto with False T \<open>k \<le> n\<close> have"(\i\n. (h (T,x) i)\<^sup>2) = T\<^sup>2 + (1 - T\<^sup>2) * (\i\{..n} - {k}. (x i)\<^sup>2 / (1 - (x k)\<^sup>2))" unfolding h_def Let_def max_def by (simp add: not_le square_le_1 power_mult_distrib power_divide if_distrib [of "\x. x ^ 2"]
sum.delta_remove sum_distrib_left) alsohave"\ = 1" using x False xk \<open>0 \<le> T\<close> by (simp add: nsphere sum_diff not_le \<open>k \<le> n\<close> power2_eq_1_iff flip: sum_divide_distrib) finallyshow ?thesis . qed moreover have"h (T,x) i = 0" if"x \ topspace (nsphere n)" "- (1/2) \ x k" and "n < i" "0 \ T" "T \ 1" for T x i proof (cases "-T \ x k ") case False thenshow ?thesis using that by (auto simp: nsphere h_def Let_def not_le max_def) qed (use that in\<open>auto simp: nsphere h\<close>) ultimatelyshow ?thesis by auto qed ultimately have cmh: "continuous_map (prod_topology ?T01 ?X12) (nsphere n) h" proof (subst (2) nsphere) qed (fastforce simp add: continuous_map_in_subtopology continuous_map_componentwise_UNIV) have"hom_induced p (subtopology (nsphere n) {x. 0 \ x k})
(topspace (subtopology (nsphere n) {x. 0 \<le> x k}) \<inter> {x. x k = 0}) ?X12
(topspace ?X12 \<inter> {x. - 1/2 \<le> x k \<and> x k \<le> 0}) id \<in> iso (relative_homology_group p (subtopology (nsphere n) {x. 0 \<le> x k})
(topspace (subtopology (nsphere n) {x. 0 \<le> x k}) \<inter> {x. x k = 0}))
(relative_homology_group p ?X12 (topspace ?X12 \<inter> {x. - 1/2 \<le> x k \<and> x k \<le> 0}))" proof (rule deformation_retract_relative_homology_group_isomorphism_id) show"retraction_maps ?X12 (subtopology (nsphere n) {x. 0 \ x k}) (h \ (\x. (0,x))) id" unfolding retraction_maps_def proof (intro conjI ballI) show"continuous_map ?X12 (subtopology (nsphere n) {x. 0 \ x k}) (h \ Pair 0)" apply (simp add: continuous_map_in_subtopology) apply (intro conjI continuous_map_compose [OF _ cmh] continuous_intros) apply (auto simp: h_def Let_def) done show"continuous_map (subtopology (nsphere n) {x. 0 \ x k}) ?X12 id" by (simp add: continuous_map_in_subtopology) qed (simp add: nsphere h) next have h0: "\xa. \xa \ topspace (nsphere n); - (1/2) \ xa k; xa k \ 0\ \ h (0, xa) k = 0" by (simp add: h_def Let_def) show"(h \ (\x. (0,x))) \ (topspace ?X12 \ {x. - 1 / 2 \ x k \ x k \ 0}) \<rightarrow> topspace (subtopology (nsphere n) {x. 0 \<le> x k}) \<inter> {x. x k = 0}" apply (auto simp: h0) apply (rule subsetD [OF continuous_map_image_subset_topspace [OF cmh]]) apply (force simp: nsphere) done have hin: "\t x. \x \ topspace (nsphere n); - (1/2) \ x k; 0 \ t; t \ 1\ \ h (t,x) \ topspace (nsphere n)" apply (rule subsetD [OF continuous_map_image_subset_topspace [OF cmh]]) apply (force simp: nsphere) done have h1: "\x. \x \ topspace (nsphere n); - (1/2) \ x k\ \ h (1, x) = x" by (simp add: h nsphere) have"continuous_map (prod_topology ?T01 ?X12) (nsphere n) h" using cmh by force thenshow"homotopic_with
(\<lambda>h. h ` (topspace ?X12 \<inter> {x. - 1 / 2 \<le> x k \<and> x k \<le> 0}) \<subseteq> topspace ?X12 \<inter> {x. - 1 / 2 \<le> x k \<and> x k \<le> 0})
?X12 ?X12 (h \<circ> (\<lambda>x. (0,x))) id" apply (subst homotopic_with, force) apply (rule_tac x=h in exI) apply (auto simp: hin h1 continuous_map_in_subtopology) apply (auto simp: h_def Let_def max_def) done qed auto thenhave 2: "hom_induced p (subtopology (nsphere n) {x. 0 \ x k}) {x. x k = 0}
?X12 {x. - 1/2 \<le> x k \<and> x k \<le> 0} id \<in> Group.iso
(relative_homology_group p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0})
(relative_homology_group p ?X12 {x. - 1/2 \<le> x k \<and> x k \<le> 0})" by (metis hom_induced_restrict relative_homology_group_restrict topspace_subtopology) show ?thesis using iso_set_trans [OF 2 1] by (simp add: subset_iff continuous_map_in_subtopology flip: hom_induced_compose) qed
corollary iso_upper_hemisphere_reduced_homology_group: "(hom_boundary (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) \ 0}) {x. x(Suc n) = 0}) \<in> iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) \<ge> 0}) {x. x(Suc n) = 0})
(reduced_homology_group p (nsphere n))" proof - have"{x. 0 \ x (Suc n)} \ {x. x (Suc n) = 0} = {x. x (Suc n) = (0::real)}" by auto thenhave n: "nsphere n = subtopology (subtopology (nsphere (Suc n)) {x. x(Suc n) \ 0}) {x. x(Suc n) = 0}" by (simp add: subtopology_nsphere_equator subtopology_subtopology) have ne: "(\i. if i = n then 1 else 0) \ topspace (subtopology (nsphere (Suc n)) {x. 0 \ x (Suc n)}) \ {x. x (Suc n) = 0}" by (simp add: nsphere if_distrib [of "\x. x ^ 2"] cong: if_cong) show ?thesis unfolding n using iso_relative_homology_of_contractible [where p = "1 + p", simplified] by (metis contractible_space_upper_hemisphere dual_order.refl empty_iff ne) qed
corollary iso_reduced_homology_group_upper_hemisphere: assumes"k \ n" shows"hom_induced p (nsphere n) {} (nsphere n) {x. x k \ 0} id \<in> iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. x k \<ge> 0})" proof (rule iso_reduced_homology_by_contractible [OF contractible_space_upper_hemisphere [OF assms]]) have"(\i. if i = k then 1 else 0) \ topspace (nsphere n) \ {x. 0 \ x k}" using assms by (simp add: nsphere if_distrib [of "\x. x ^ 2"] cong: if_cong) thenshow"topspace (nsphere n) \ {x. 0 \ x k} \ {}" by blast qed
lemma iso_relative_homology_group_lower_hemisphere: "hom_induced p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0} (nsphere n) {x. x k \ 0} id \<in> iso (relative_homology_group p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0})
(relative_homology_group p (nsphere n) {x. x k \<ge> 0})" (is "?k \<in> iso ?G ?H") proof -
define r where"r \ \x i. if i = k then -x i else (x i::real)" thenhave [simp]: "r \ r = id" by force have cmr: "continuous_map (subtopology (nsphere n) S) (nsphere n) r"for S using continuous_map_nsphere_reflection [of n k] by (simp add: continuous_map_from_subtopology r_def) let ?f = "hom_induced p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0}
(subtopology (nsphere n) {x. x k \<ge> 0}) {x. x k = 0} r" let ?g = "hom_induced p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0} (nsphere n) {x. x k \ 0} id" let ?h = "hom_induced p (nsphere n) {x. x k \ 0} (nsphere n) {x. x k \ 0} r" obtain f h where
f: "f \ iso ?G (relative_homology_group p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0})" and h: "h \ iso (relative_homology_group p (nsphere n) {x. x k \ 0}) ?H" and eq: "h \ ?g \ f = ?k" proof have hmr: "homeomorphic_map (nsphere n) (nsphere n) r" unfolding homeomorphic_map_maps by (metis \<open>r \<circ> r = id\<close> cmr homeomorphic_maps_involution pointfree_idE subtopology_topspace) thenhave hmrs: "homeomorphic_map (subtopology (nsphere n) {x. x k \ 0}) (subtopology (nsphere n) {x. x k \ 0}) r" by (simp add: homeomorphic_map_subtopologies_alt r_def) have rimeq: "r ` (topspace (subtopology (nsphere n) {x. x k \ 0}) \ {x. x k = 0})
= topspace (subtopology (nsphere n) {x. 0 \<le> x k}) \<inter> {x. x k = 0}" using continuous_map_eq_topcontinuous_at continuous_map_nsphere_reflection topcontinuous_at_atin by (fastforce simp: r_def Pi_iff) show"?f \ iso ?G (relative_homology_group p (subtopology (nsphere n) {x. x k \ 0}) {x. x k = 0})" using homeomorphic_map_relative_homology_iso [OF hmrs Int_lower1 rimeq] by (metis hom_induced_restrict relative_homology_group_restrict) have rimeq: "r ` (topspace (nsphere n) \ {x. x k \ 0}) = topspace (nsphere n) \ {x. 0 \ x k}" by (metis hmrs homeomorphic_imp_surjective_map topspace_subtopology) show"?h \ Group.iso (relative_homology_group p (nsphere n) {x. x k \ 0}) ?H" using homeomorphic_map_relative_homology_iso [OF hmr Int_lower1 rimeq] by simp have [simp]: "\x. x k = 0 \ r x k = 0" by (auto simp: r_def) have"?h \ ?g \ ?f
= hom_induced p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0} (nsphere n) {x. 0 \<le> x k} r \<circ>
hom_induced p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0} (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0} r" apply (subst hom_induced_compose [symmetric]) using continuous_map_nsphere_reflection apply (force simp: r_def)+ done alsohave"\ = ?k" apply (subst hom_induced_compose [symmetric]) apply (simp_all add: image_subset_iff cmr) using hmrs homeomorphic_imp_continuous_map apply blast done finallyshow"?h \ ?g \ ?f = ?k" . qed with iso_relative_homology_group_upper_hemisphere [of p n k] have"h \ hom_induced p (subtopology (nsphere n) {f. 0 \ f k}) {f. f k = 0} (nsphere n) {f. f k \ 0} id \ f \<in> Group.iso ?G (relative_homology_group p (nsphere n) {f. 0 \<le> f k})" using f h iso_set_trans by blast thenshow ?thesis by (simp add: eq) qed
lemma iso_lower_hemisphere_reduced_homology_group: "hom_boundary (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) \ 0}) {x. x(Suc n) = 0} \<in> iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) \<le> 0})
{x. x(Suc n) = 0})
(reduced_homology_group p (nsphere n))" proof - have"{x. (\i\n. (x i)\<^sup>2) = 1 \ (\i>n. x i = 0)} =
({x. (\<Sum>i\<le>n. (x i)\<^sup>2) + (x (Suc n))\<^sup>2 = 1 \<and> (\<forall>i>Suc n. x i = 0)} \<inter> {x. x (Suc n) \<le> 0} \<inter>
{x. x (Suc n) = (0::real)})" by (force simp: dest: Suc_lessI) thenhave n: "nsphere n = subtopology (subtopology (nsphere (Suc n)) {x. x(Suc n) \ 0}) {x. x(Suc n) = 0}" by (simp add: nsphere subtopology_subtopology) have ne: "(\i. if i = n then 1 else 0) \ topspace (subtopology (nsphere (Suc n)) {x. x (Suc n) \ 0}) \ {x. x (Suc n) = 0}" by (simp add: nsphere if_distrib [of "\x. x ^ 2"] cong: if_cong) show ?thesis unfolding n apply (rule iso_relative_homology_of_contractible [where p = "1 + p", simplified]) using contractible_space_lower_hemisphere ne apply blast+ done qed
lemma isomorphism_sym: "\f \ iso G1 G2; \x. x \ carrier G1 \ r'(f x) = f(r x); \<And>x. x \<in> carrier G1 \<Longrightarrow> r x \<in> carrier G1; group G1; group G2\<rbrakk> \<Longrightarrow> \<exists>f \<in> iso G2 G1. \<forall>x \<in> carrier G2. r(f x) = f(r' x)" apply (clarsimp simp add: group.iso_iff_group_isomorphisms Bex_def) by (metis (full_types) group_isomorphisms_def group_isomorphisms_sym hom_in_carrier)
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