# Theory Brouwer_Degree

```section‹Homology, III: Brouwer Degree›

theory Brouwer_Degree
imports Homology_Groups "HOL-Algebra.Multiplicative_Group"

begin

subsection‹Reduced Homology›

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 {()}) {} (λx. ())))"

lemma one_reduced_homology_group: "𝟭⇘reduced_homology_group p X⇙ = 𝟭⇘homology_group p X⇙"

lemma group_reduced_homology_group [simp]: "group (reduced_homology_group p X)"

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 {()}) {} (λx. ()))"
(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)"

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 {()}) {} (λx. ())))
= 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 {()}) {} (λx. ()))"
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
qed

lemma trivial_reduced_homology_group:
"p < 0 ⟹ trivial_group(reduced_homology_group p X)"

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
⟹ hom_induced p X {} (discrete_topology {()}) {} (λx. ())
= (hom_induced p Y {} (discrete_topology {()}) {} (λx. ()) ∘ 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 {()}) {} (λx. ()))"
"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 = (λc. one(reduced_homology_group p Y))"
by (force simp: hom_induced_default reduced_homology_group_def)
then show ?thesis
qed

lemma hom_induced_reduced:
"c ∈ carrier(reduced_homology_group p X)
⟹ hom_induced p X {} Y {} f c ∈ 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
∈ hom (relative_homology_group p X S) (reduced_homology_group (p-1) (subtopology X S))"
proof -
have *: "continuous_map X (discrete_topology {()}) (λx. ())" "(λx. ()) ` 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)
⊆ kernel (homology_group (p - 1) (subtopology X S))
(homology_group (p - 1) (discrete_topology {()}))
(hom_induced (p - 1) (subtopology X S) {}
(discrete_topology {()}) {} (λx. ()))"
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 {()}) {()} (λx. ()) c)
= 𝟭⇘homology_group (p - 1) (discrete_topology {()})⇙"
by (metis hom_induced_carrier local.hom_one singletonD triv trivial_group_def)
then show "hom_induced (p - 1) (subtopology X S) {} (discrete_topology {()}) {} (λx. ()) (hom_boundary p X S c) =
𝟭⇘homology_group (p - 1) (discrete_topology {()})⇙"
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 (λh. True) X X (g ∘ f) id"
and fg: "homotopic_with (λk. 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"
qed

lemma homotopy_equivalence_reduced_homology_group_isomorphism:
assumes "continuous_map X Y f" "continuous_map Y X g"
and "homotopic_with (λh. True) X X (g ∘ f) id" "homotopic_with (λk. True) Y Y (f ∘ g) id"
shows "(hom_induced p X {} Y {} f)
∈ 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)"
qed

lemma homotopy_equivalent_space_imp_isomorphic_reduced_homology_groups:
"X homotopy_equivalent_space Y
⟹ reduced_homology_group p X ≅ 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"

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 {()}) {} (λx. ()))
∈ 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 ‹auto simp: iso_kernel_image›)
qed

lemma trivial_reduced_homology_group_contractible_space:
"contractible_space X ⟹ trivial_group (reduced_homology_group p X)"
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 {()}) {} (λx. ()) x'
= 𝟭⇘homology_group p (discrete_topology {()})⇙"
if "y ∈ carrier ?H" for y
proof -
let ?f = "hom_induced p (discrete_topology {()}) {} X {} (λx. a)"
let ?g = "hom_induced p X {} (discrete_topology {()}) {} (λx. ())"
have bcarr: "?f (?g y) ∈ carrier ?H"
interpret gh1:
group_hom "relative_homology_group p X S" "relative_homology_group p (discrete_topology {()}) {()}"
"hom_induced p X S (discrete_topology {()}) {()} (λx. ())"
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 (λx. 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 (λx. a) ∘
hom_induced p (discrete_topology {()}) {} (discrete_topology {()}) {()} id ∘ ?g) y"
by (simp add: a ‹a ∈ S› flip: hom_induced_compose)
also have "… = 𝟭⇘relative_homology_group p X S⇙"
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)) = 𝟭⇘relative_homology_group p X S⇙"
by simp
then show "?h y = ?h (y ⊗⇘?H⇙ inv⇘?H⇙ ?f (?g y))"
show "(y ⊗⇘?H⇙ inv⇘?H⇙ ?f (?g y)) ∈ carrier (homology_group p X)"
have *: "(?g ∘ hom_induced p X {} X {} (λx. a)) y = hom_induced p X {} (discrete_topology {()}) {} (λa. ()) y"
by (simp add: a ‹a ∈ S› flip: hom_induced_compose)
have "?g (y ⊗⇘?H⇙ inv⇘?H⇙ (?f ∘ ?g) y)
= 𝟭⇘homology_group p (discrete_topology {()})⇙"
by (simp add: a ‹a ∈ S› that hom_induced_carrier flip: hom_induced_compose * [unfolded o_def])
then show "?g (y ⊗⇘?H⇙ inv⇘?H⇙ ?f (?g y))
= 𝟭⇘homology_group p (discrete_topology {()})⇙"
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])
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
⊆ (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))
⊆ (hom_induced p (subtopology X S) {} X {} id) ` carrier ?G3"
proof
show "?h2 ` carrier ?G3 ⊆ carrier ?G2"
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 {()}) {} (λx. ())))
(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))
⊆ ?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
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‹More homology properties of deformations, retracts, contractible spaces›

lemma iso_relative_homology_of_contractible:
"⟦contractible_space X; topspace X ∩ S ≠ {}⟧
⟹ hom_boundary p X S
∈ 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 ≠ {}⟧
⟹ relative_homology_group p X S ≅
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 ≠ {}⟧
⟹ reduced_homology_group p (subtopology X S) ≅ relative_homology_group(p + 1) X S"

lemma iso_reduced_homology_by_contractible:
"⟦contractible_space(subtopology X S); topspace X ∩ S ≠ {}⟧
⟹ (hom_induced p X {} X S id) ∈ 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 ≠ {}⟧
⟹ reduced_homology_group p X ≅ 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 ≠ {}⟧
⟹ relative_homology_group p X S ≅ 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})"

lemma isomorphic_relative_homology_by_singleton:
"a ∈ topspace X ⟹ relative_homology_group p X ({a}) ≅ reduced_homology_group p X"

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 (λh. h ` U ⊆ U) X X (s ∘ r) id⟧
⟹ 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 (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 (λh. h ` U ⊆ U) X X r id⟧
⟹ 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)"

lemma deformation_retract_relative_homology_group_isomorphism:
"⟦retraction_maps X Y r id; V ⊆ U; r ` U ⊆ V; homotopic_with (λh. h ` U ⊆ U) X X r id⟧
⟹ (hom_induced p X U Y V r) ∈ 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 (λh. h ` U ⊆ U) X X r id⟧
⟹ (hom_induced p Y V X U id) ∈ 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 (λh. h ` U ⊆ U) X X (s ∘ r) id⟧
⟹ relative_homology_group p X U ≅ 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 (λh. True) X X (s ∘ r) id⟧
⟹ homology_group p X ≅ homology_group p Y"

lemma deformation_retract_imp_isomorphic_relative_homology_groups:
"⟦retraction_maps X X' r id; V ⊆ U; r ` U ⊆ V; homotopic_with (λh. h ` U ⊆ U) X X r id⟧
⟹ relative_homology_group p X U ≅ relative_homology_group p X' V"

lemma deformation_retract_imp_isomorphic_homology_groups:
"⟦retraction_maps X X' r id; homotopic_with (λh. True) X X r id⟧
⟹ homology_group p X ≅ homology_group p X'"

lemma epi_hom_induced_inclusion:
assumes "homotopic_with (λx. True) X X id f" and "f ` (topspace X) ⊆ S"
shows "(hom_induced p (subtopology X S) {} X {} id)
∈ 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
∈ hom (homology_group p (subtopology X S)) (homology_group p X)"
show "hom_induced p X {} (subtopology X S) {} f
∈ carrier (homology_group p X) → carrier (homology_group p (subtopology X S))"
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 (λx. 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)
∈ 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 (λx. 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)
∈ 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, 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 (λx. 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 (λx. 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))"
"(λ(x, y). x ⊗⇘homology_group p (subtopology X S)⇙ y)
∈ Group.iso (subgroup_generated (homology_group p (subtopology X S)) H ××
subgroup_generated (homology_group p (subtopology X S)) K)
(homology_group p (subtopology X S))"
"hom_boundary (p + 1) X S
∈ 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
∈ 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 (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 "(λ(x, y). x ⊗⇘?pXS⇙ y)
∈ Group.iso (subgroup_generated (?pXS) H ×× 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)"
qed
qed

lemma iso_homology_group_prod_deformation:
assumes "homotopic_with (λx. True) X X id f" and "f ` (topspace X) ⊆ S"
shows "homology_group p (subtopology X S)
≅ DirProd (homology_group p X) (relative_homology_group(p + 1) X S)"
(is "?G ≅ DirProd ?H ?R")
proof -
obtain H K where HK:
"(λ(x, y). x ⊗⇘?G⇙ y)
∈ Group.iso (subgroup_generated (?G) H ×× 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"
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 (λx. 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 ≠ {}⟧
⟹ homology_group p (subtopology X S) ≅ 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 ≠ {}⟧
⟹ 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 (λk. 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 (λh. 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 (λk. 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 ‹T ⊆ S›] homology_exactness_triple_3 [OF ‹T ⊆ S›]
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)
∈ 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 ‹T ⊆ S›] homology_exactness_triple_3 [OF ‹T ⊆ S›]
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
∈ 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 ‹T ⊆ S›] homology_exactness_triple_2 [OF ‹T ⊆ S›] 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 {})"
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 ‹T ≠ {}›
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 ‹T ≠ {}› 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 ‹T ≠ {}›
also have "…  ≅ relative_homology_group (p+1) Y T"
also have "… ≅ homology_group p (subtopology Y T)"
by (simp add: False ‹T ≠ {}› assms group.iso_sym iso_homology_contractible_space_subtopology2)
finally show ?thesis .
qed
qed

subsection‹Homology groups of spheres›

lemma iso_reduced_homology_group_lower_hemisphere:
assumes "k ≤ n"
shows "hom_induced p (nsphere n) {} (nsphere n) {x. x k ≤ 0} id
∈ iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. x k ≤ 0})"
proof (rule iso_reduced_homology_by_contractible)
show "contractible_space (subtopology (nsphere n) {x. x k ≤ 0})"
have "(λi. if i = k then -1 else 0) ∈ topspace (nsphere n) ∩ {x. x k ≤ 0}"
using assms by (simp add: nsphere if_distrib [of "λx. 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)⇧2 ≤ 1"
proof (cases "k ≤ n")
case True
have "(∑i ∈ {..n} - {k}. (x i)⇧2) = (∑i≤n. (x i)⇧2) - (x k)⇧2"
using ‹k ≤ n› by (simp add: sum_diff)
then show ?thesis
using assms
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)⇧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)⇧2) = (∑i≤n. (x i)⇧2) - (x k)⇧2"
using ‹k ≤ n› by (simp add: sum_diff)
also have "… = 0"
using assms by (simp add: nsphere)
finally have "∀i∈{..n} - {k}. (x i)⇧2 = 0"
then show ?thesis
using True ‹i ≠ k› by auto
next
case False
with x show ?thesis
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)
∈ iso (relative_homology_group p (subtopology (nsphere n) {x. x k ≥ 0}) {x. x k = 0})
(relative_homology_group p (nsphere n) {x. x k ≤ 0})" (is "?h ∈ 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 "λx::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 "λx::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
∈ iso (relative_homology_group p ?X12 {x. -1/2 ≤ x k ∧ x k ≤ 0})
?H"
using homology_excision_axiom [OF nn subset_UNIV, of p] by simp
define h where "h ≡ λ(T,x). let y = max (x k) (-T) in
(λi. 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)⇧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 (λx. h x i)" for i
proof -
show ?thesis
proof (rule continuous_map_eq)
show "continuous_map (prod_topology ?T01 ?X12)
euclideanreal (λ(T, x). if 0 ≤ 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 (λx. snd x k)"
apply (subst continuous_map_of_snd [unfolded o_def])
next
show "continuous_map (subtopology (prod_topology ?T01 ?X12) {p ∈ topspace (prod_topology ?T01 ?X12). 0 ≤ snd p k})
euclideanreal (λx. snd x i)"
apply (rule continuous_map_from_subtopology)
apply (subst continuous_map_of_snd [unfolded o_def])
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 (λx. snd x k)" for k S T
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 (λx. 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}))
⊆ {x. (∑i≤n. (x i)⇧2) = 1 ∧ (∀i>n. x i = 0)}"
proof -
have "(∑i≤n. (h (T,x) i)⇧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 ‹0 ≤ T› have "k ≤ n"
by (metis neg_le_0_iff_le not_le)
have "1 - (x k)⇧2 ≥ 0"
using topspace_nsphere_1 x by auto
with False T ‹k ≤ n›
have "(∑i≤n. (h (T,x) i)⇧2) = T⇧2 + (1 - T⇧2) * (∑i∈{..n} - {k}. (x i)⇧2 / (1 - (x k)⇧2))"
unfolding h_def Let_def max_def
by (simp add: not_le square_le_1 power_mult_distrib power_divide if_distrib [of "λx. x ^ 2"]
sum.delta_remove sum_distrib_left)
also have "… = 1"
using x False xk ‹0 ≤ T›
by (simp add: nsphere sum_diff not_le ‹k ≤ n› 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 ‹auto simp: nsphere h›)
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 ≤ x k}) ∩ {x. x k = 0}) ?X12
(topspace ?X12 ∩ {x. - 1/2 ≤ x k ∧ x k ≤ 0}) id
∈ iso (relative_homology_group p (subtopology (nsphere n) {x. 0 ≤ x k})
(topspace (subtopology (nsphere n) {x. 0 ≤ x k}) ∩ {x. x k = 0}))
(relative_homology_group p ?X12 (topspace ?X12 ∩ {x. - 1/2 ≤ x k ∧ x k ≤ 0}))"
proof (rule deformation_retract_relative_homology_group_isomorphism_id)
show "retraction_maps ?X12 (subtopology (nsphere n) {x. 0 ≤ x k}) (h ∘ (λx. (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 (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)
next
have h0: "⋀xa. ⟦xa ∈ topspace (nsphere n); - (1/2) ≤ xa k; xa k ≤ 0⟧ ⟹ h (0, xa) k = 0"
show "(h ∘ (λx. (0,x))) ` (topspace ?X12 ∩ {x. - 1 / 2 ≤ x k ∧ x k ≤ 0})
⊆ topspace (subtopology (nsphere n) {x. 0 ≤ x k}) ∩ {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"
have "continuous_map (prod_topology ?T01 ?X12) (nsphere n) h"
using cmh by force
then show "homotopic_with
(λh. h ` (topspace ?X12 ∩ {x. - 1 / 2 ≤ x k ∧ x k ≤ 0}) ⊆ topspace ?X12 ∩ {x. - 1 / 2 ≤ x k ∧ x k ≤ 0})
?X12 ?X12 (h ∘ (λx. (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 ≤ x k ∧ x k ≤ 0} id
∈ Group.iso
(relative_homology_group p (subtopology (nsphere n) {x. 0 ≤ x k}) {x. x k = 0})
(relative_homology_group p ?X12 {x. - 1/2 ≤ x k ∧ x k ≤ 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})
∈ iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) ≥ 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}"
have ne: "(λi. 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 "λx. 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
∈ iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. x k ≥ 0})"
proof (rule iso_reduced_homology_by_contractible [OF contractible_space_upper_hemisphere [OF assms]])
have "(λi. if i = k then 1 else 0) ∈ topspace (nsphere n) ∩ {x. 0 ≤ x k}"
using assms by (simp add: nsphere if_distrib [of "λx. 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
∈ iso (relative_homology_group p (subtopology (nsphere n) {x. x k ≤ 0}) {x. x k = 0})
(relative_homology_group p (nsphere n) {x. x k ≥ 0})" (is "?k ∈ iso ?G ?H")
proof -
define r where "r ≡ λx 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]
let ?f = "hom_induced p (subtopology (nsphere n) {x. x k ≤ 0}) {x. x k = 0}
(subtopology (nsphere n) {x. x k ≥ 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 ‹r ∘ r = id› 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"
have rimeq: "r ` (topspace (subtopology (nsphere n) {x. x k ≤ 0}) ∩ {x. x k = 0})
= topspace (subtopology (nsphere n) {x. 0 ≤ x k}) ∩ {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 ≤ x k}) {x. x k = 0} (nsphere n) {x. 0 ≤ x k} r ∘
hom_induced p (subtopology (nsphere n) {x. x k ≤ 0}) {x. x k = 0} (subtopology (nsphere n) {x. 0 ≤ 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])
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
∈ Group.iso ?G (relative_homology_group p (nsphere n) {f. 0 ≤ f k})"
using f h iso_set_trans by blast
then show ?thesis
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}
∈ iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. x(Suc n) ≤ 0})
{x. x(Suc n) = 0})
(reduced_homology_group p (nsphere n))"
proof -
have "{x. (∑i≤n. (x i)⇧2) = 1 ∧ (∀i>n. x i = 0)} =
({x. (∑i≤n. (x i)⇧2) + (x (Suc n))⇧2 = 1 ∧ (∀i>Suc n. x i = 0)} ∩ {x. x (Suc n) ≤ 0} ∩
{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}"
have ne: "(λi. 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 "λx. 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);
⋀x. x ∈ carrier G1 ⟹ r x ∈ carrier G1; group G1; group G2⟧
⟹ ∃f ∈ iso G2 G1. ∀x ∈ 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)⟧
⟹ ∃f ∈ iso G1 G3. ∀x ∈ 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))).
∀c ∈ carrier(reduced_homology_group p (nsphere n)).
hom_induced (1 + p) (nsphere(Suc n)) {} (nsphere(Suc n)) {}
(λx i. if i = 0 then -x i else x i) (f c)
= f (hom_induced p (nsphere n) {} (nsphere n) {} (λx i. if i = 0 then -x i else x i) c)"
proof -
define r where "r ≡ λx::nat⇒real. λi. 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 = "λp 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.
∀c∈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
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} ∘
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)
show "∃f∈Group.iso ?G2 ?H2.
∀c∈carrier ?G2.
?s2 (f c) = f (hom_induced (1 + p) ?sub {x. x (Suc n) = 0} ?sub {x. x (Suc n) = 0} r c)"
proof (intro ballI bexI)
fix c
assume "c ∈ carrier (relative_homology_group (1 + p) ?sub {x. x (Suc n) = 0})"
show "?s2 (hom_induced (1 + p) ?sub {x. x (Suc n) = 0} (nsphere (Suc n)) {x. x (Suc n) ≤ 0} id c)
= hom_induced (1 + p) ?sub {x. x (Suc n) = 0} (nsphere (Suc n)) {x. x (Suc n) ≤ 0} id (?r2 c)"
apply (simp add: rsub hom_induced_compose' Collect_mono_iff cmr)
apply (subst hom_induced_compose')
apply (simp_all add: continuous_map_in_subtopology continuous_map_from_subtopology [OF cmr] rsub)
apply (auto simp: r_def)
done
next
let ?h = "hom_induced (1 + p) (nsphere(Suc n)) {} (nsphere (Suc n)) {x. x(Suc n) ≤ 0} id"
show "∃f∈Group.iso ?H2 (reduced_homology_group (1 + p) (nsphere (Suc n))).
∀c∈carrier ?H2. ?j (1 + p) (Suc n) (f c) = f (?s2 c)"
proof (rule isomorphism_sym)
show "?h ∈ Group.iso (reduced_homology_group (1 + p) (nsphere (Suc n)))
(relative_homology_group (1 + p) (nsphere (Suc n)) {x. x (Suc n) ≤ 0})"
using iso_reduced_homology_group_lower_hemisphere by blast
next
fix c
assume "c ∈ carrier (reduced_homology_group (1 + p) (nsphere (Suc n)))"
show "?s2 (?h c) = ?h (?j (1 + p) (Suc n)  c)"
by (simp add: hom_induced_compose' cmr rsub)
next
fix c
assume "c ∈ carrier (reduced_homology_group (1 + p) (nsphere (Suc n)))"
then show "hom_induced (1 + p) (nsphere (Suc n)) {} (nsphere (Suc n)) {} r c
∈ carrier (reduced_homology_group (1 + p) (nsphere (Suc n)))"
qed auto
qed
qed
qed

lemma reduced_homology_group_nsphere_aux:
"if p = int n then reduced_homology_group n (nsphere n) ≅ integer_group
else trivial_group(reduced_homology_group p (nsphere n))"
proof (induction n arbitrary: p)
case 0
let ?a = "λi::nat. if i = 0 then 1 else (0::real)"
let ?b = "λi::nat. if i = 0 then -1 else (0::real)"
have st: "subtopology (powertop_real UNIV) {?a, ?b} = nsphere 0"
proof -
have "{?a, ?b} ```