section\<open>Homology, III: Brouwer Degree\<close>
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
then show ?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
then have "hom_induced p X {} Y {} f = (\c. one(reduced_homology_group p Y))"
by (force simp: hom_induced_default reduced_homology_group_def)
then show ?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)
then show "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
then show ?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)"
then have "(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
then show "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)"
then have "(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
then show "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)
also have "\ = \\<^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)
finally have "?h (?f (?g y)) = \\<^bsub>relative_homology_group p X S\<^esub>"
by simp
then show "?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])
then show "?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)
then show "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
then show ?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)
also have "\ \ ?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
finally show ?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 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)
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 \ carrier (homology_group p X)"
then show "hom_induced p (subtopology X S) {} X {} id (hom_induced p X {} (subtopology X S) {} f x) = x"
by (metis assms continuous_map_id_subt continuous_map_in_subtopology hom_induced_compose' hom_induced_id homology_homotopy_empty homotopic_with_imp_continuous_maps image_empty order_refl)
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)
then show ?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)
then show ?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, hide_lams) 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 (meson assms continuous_map_in_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"
apply (rule splitting_lemma_right [OF x, where g' = "hom_induced p X {} (subtopology X S) {} f"])
apply (simp add: hom_induced_empty_hom)
apply (simp add: contf hom_induced_compose')
apply (metis (full_types) assms(1) hom_induced_id homology_homotopy_empty)
apply blast
done
show ?thesis
proof
show "subgroup H ?pXS"
using HK(1) normal_imp_subgroup by blast
then show "(\(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)
also have "\ \ DirProd ?R ?H"
by (meson HK group.DirProd_iso_trans group.group_subgroup_generated group.iso_sym group_relative_homology_group is_isoI)
also have "\ \ DirProd ?H ?R"
by (simp add: DirProd_commute_iso)
finally show ?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
then have "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
also have "\ \ 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
finally show ?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, hide_lams) 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)
then show ?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)
also have "\ \ integer_group \\ relative_homology_group 1 Y T"
by (simp add: assms group.DirProd_iso_trans iso_refl)
also have "\ \ homology_group p (subtopology Y T)"
by (simp add: True \<open>T \<noteq> {}\<close> assms group.iso_sym iso_homology_contractible_space_subtopology1)
finally show ?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)
also have "\ \ relative_homology_group (p+1) Y T"
by (simp add: assms)
also have "\ \ homology_group p (subtopology Y T)"
by (simp add: False \<open>T \<noteq> {}\<close> assms group.iso_sym iso_homology_contractible_space_subtopology2)
finally show ?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)
then show "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)
then show ?thesis
using assms
apply (simp add: nsphere)
by (metis diff_ge_0_iff_ge sum_nonneg zero_le_power2)
next
case False
then show ?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)
also have "\ = 0"
using assms by (simp add: nsphere)
finally have "\i\{..n} - {k}. (x i)\<^sup>2 = 0"
by (simp add: sum_nonneg_eq_0_iff)
then show ?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
moreover have "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
ultimately have "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)
also have "\ \ {x \ topspace (nsphere n). x k \ {y. y < 0}}"
by force
also have "\ \ 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
finally have 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
then show ?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)
also have "\ = 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)
finally show ?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
then show ?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>)
ultimately show ?thesis
by auto
qed
ultimately
have cmh: "continuous_map (prod_topology ?T01 ?X12) (nsphere n) h"
by (subst (2) nsphere) (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) (auto simp: nsphere)
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})
\<subseteq> 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
then show "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
then have 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
then have 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
apply (rule iso_relative_homology_of_contractible [where p = "1 + p", simplified])
using contractible_space_upper_hemisphere ne apply blast+
done
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)
then show "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)"
then have [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)
then have 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)
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
also have "\ = ?k"
apply (subst hom_induced_compose [symmetric])
apply (simp_all add: image_subset_iff cmr)
using hmrs homeomorphic_imp_continuous_map apply blast
done
finally show "?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
then show ?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)
then have 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)
lemma isomorphism_trans:
"\\f \ iso G1 G2. \x \ carrier G1. r2(f x) = f(r1 x); \f \ iso G2 G3. \x \ carrier G2. r3(f x) = f(r2 x)\
\<Longrightarrow> \<exists>f \<in> iso G1 G3. \<forall>x \<in> carrier G1. r3(f x) = f(r1 x)"
apply clarify
apply (rename_tac g f)
apply (rule_tac x="f \ g" in bexI)
apply (metis iso_iff comp_apply hom_in_carrier)
using iso_set_trans by blast
lemma reduced_homology_group_nsphere_step:
"\f \ iso(reduced_homology_group p (nsphere n))
(reduced_homology_group (1 + p) (nsphere (Suc n))).
\<forall>c \<in> carrier(reduced_homology_group p (nsphere n)).
hom_induced (1 + p) (nsphere(Suc n)) {} (nsphere(Suc n)) {}
(\<lambda>x i. if i = 0 then -x i else x i) (f c)
= f (hom_induced p (nsphere n) {} (nsphere n) {} (\<lambda>x i. if i = 0 then -x i else x i) c)"
proof -
define r where "r \ \x::nat\real. \i. if i = 0 then -x i else x i"
have cmr: "continuous_map (nsphere n) (nsphere n) r" for n
unfolding r_def by (rule continuous_map_nsphere_reflection)
have rsub: "r ` {x. 0 \ x (Suc n)} \ {x. 0 \ x (Suc n)}" "r ` {x. x (Suc n) \ 0} \ {x. x (Suc n) \ 0}" "r ` {x. x (Suc n) = 0} \ {x. x (Suc n) = 0}"
by (force simp: r_def)+
let ?sub = "subtopology (nsphere (Suc n)) {x. x (Suc n) \ 0}"
let ?G2 = "relative_homology_group (1 + p) ?sub {x. x (Suc n) = 0}"
let ?r2 = "hom_induced (1 + p) ?sub {x. x (Suc n) = 0} ?sub {x. x (Suc n) = 0} r"
let ?j = "\p n. hom_induced p (nsphere n) {} (nsphere n) {} r"
show ?thesis
unfolding r_def [symmetric]
proof (rule isomorphism_trans)
let ?f = "hom_boundary (1 + p) ?sub {x. x (Suc n) = 0}"
show "\f\Group.iso (reduced_homology_group p (nsphere n)) ?G2.
\<forall>c\<in>carrier (reduced_homology_group p (nsphere n)). ?r2 (f c) = f (?j p n c)"
proof (rule isomorphism_sym)
show "?f \ Group.iso ?G2 (reduced_homology_group p (nsphere n))"
using iso_upper_hemisphere_reduced_homology_group
by (metis add.commute)
next
fix c
assume "c \ carrier ?G2"
have cmrs: "continuous_map ?sub ?sub r"
by (metis (mono_tags, lifting) IntE cmr continuous_map_from_subtopology continuous_map_in_subtopology image_subset_iff rsub(1) topspace_subtopology)
have "hom_induced p (nsphere n) {} (nsphere n) {} r \ hom_boundary (1 + p) ?sub {x. x (Suc n) = 0}
= hom_boundary (1 + p) ?sub {x. x (Suc n) = 0} \<circ>
hom_induced (1 + p) ?sub {x. x (Suc n) = 0} ?sub {x. x (Suc n) = 0} r"
using naturality_hom_induced [OF cmrs rsub(3), symmetric, of "1+p", simplified]
by (simp add: subtopology_subtopology subtopology_nsphere_equator flip: Collect_conj_eq cong: rev_conj_cong)
then show "?j p n (?f c) = ?f (hom_induced (1 + p) ?sub {x. x (Suc n) = 0} ?sub {x. x (Suc n) = 0} r c)"
by (metis comp_def)
next
fix c
assume "c \ carrier ?G2"
show "hom_induced (1 + p) ?sub {x. x (Suc n) = 0} ?sub {x. x (Suc n) = 0} r c \ carrier ?G2"
using hom_induced_carrier by blast
qed auto
next
let ?H2 = "relative_homology_group (1 + p) (nsphere (Suc n)) {x. x (Suc n) \ 0}"
let ?s2 = "hom_induced (1 + p) (nsphere (Suc n)) {x. x (Suc n) \ 0} (nsphere (Suc n)) {x. x (Suc n) \ 0} r"
show "\f\Group.iso ?G2 (reduced_homology_group (1 + p) (nsphere (Suc n))). \c\carrier ?G2. ?j (1 + p) (Suc n) (f c)
= f (?r2 c)"
proof (rule isomorphism_trans)
--> --------------------
--> maximum size reached
--> --------------------
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