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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
namespace OAI
namespace BorsukNine
open Set
open scoped Topology
lemma compact_local_homotopy {X E : Type*} [TopologicalSpace X] [T2Space X]
[LocallyCompactSpace X] [TopologicalSpace E] [Zero E]
{U C : Set X} (hU : IsOpen U) (hC : IsCompact C) (hCU : C ⊆ U)
(H : X → E) (hH : Continuous H) (hHz : ∀ x, H x ≠ 0)
(K : ℝ × X → E) (hK : ContinuousOn K (Icc (0:ℝ) 1 ×ˢ U))
(hK0 : ∀ x ∈ U, K (0,x) = H x)
(hKz : ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ U, K (t,x) ≠ 0)
(B : Set E) (houtside : ∀ x ∉ U, H x ∉ B)
(hhits : ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ U, K (t,x) ∈ B → x ∈ C)
(hend : ∀ x ∈ U, K (1,x) ∉ B) :
∃ G : ℝ × X → E, ContinuousOn G (Icc (0:ℝ) 1 ×ˢ Set.univ) ∧
(∀ x, G (0,x) = H x) ∧
(∀ t ∈ Icc (0:ℝ) 1, ∀ x, G (t,x) ≠ 0) ∧
(∀ x, G (1,x) ∉ B) := by
classical
obtain ⟨q,hq1,-,hqs,hqb⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hC hU hCU
let G : ℝ × X → E := fun p => if p.2 ∈ U then K (p.1 * q p.2,p.2) else H p.2
have htm (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (x : X) : t * q x ∈ Icc (0:ℝ) 1 :=
⟨mul_nonneg ht.1 (hqb x).1,
(mul_le_mul_of_nonneg_right ht.2 (hqb x).1).trans (by simpa using (hqb x).2)⟩
have hG : ContinuousOn G (Icc (0:ℝ) 1 ×ˢ Set.univ) := by
intro p hp
by_cases hx : p.2 ∈ U
· have hc : ContinuousOn (fun p : ℝ × X => K (p.1 * q p.2,p.2))
(Icc (0:ℝ) 1 ×ˢ U) := hK.comp
((continuous_fst.mul (q.continuous.comp continuous_snd)).prodMk continuous_snd).continuousOn
(fun p hp => ⟨htm p.1 hp.1 p.2,hp.2⟩)
have hnear : ∀ᶠ p' in 𝓝 p, p'.2 ∈ U :=
continuous_snd.continuousAt.preimage_mem_nhds (hU.mem_nhds hx)
have hset : Icc (0:ℝ) 1 ×ˢ U ∈ 𝓝[Icc (0:ℝ) 1 ×ˢ Set.univ] p := by
filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear,
self_mem_nhdsWithin] with p' hu hi
exact ⟨hi.1,hu⟩
apply ((hc p ⟨hp.1,hx⟩).mono_of_mem_nhdsWithin hset).congr_of_eventuallyEq
· filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear] with p' hu
simp [G,hu]
· simp [G,hx]
· have hxq : p.2 ∉ tsupport q := fun h => hx (hqs h)
have hnear : ∀ᶠ p' in 𝓝 p, p'.2 ∉ tsupport q :=
continuous_snd.continuousAt.preimage_mem_nhds ((isClosed_tsupport q).isOpen_compl.mem_nhds hxq)
apply (hH.comp continuous_snd).continuousAt.continuousWithinAt.congr_of_eventuallyEq
· filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear] with p' hq
have hz : q p'.2 = 0 := image_eq_zero_of_notMem_tsupport hq
by_cases hu : p'.2 ∈ U
· simp [G,hu,hz,hK0 p'.2 hu]
· simp [G,hu]
· simp [G,hx]
refine ⟨G,hG,?_,?_,?_⟩
· intro x
by_cases hx : x ∈ U
· simp [G,hx,hK0 x hx]
· simp [G,hx]
· intro t ht x
by_cases hx : x ∈ U
· simpa only [G,ite_eq_left hx] using hKz _ (htm t ht x) x hx
· simpa only [G,ite_eq_right hx] using hHz x
· intro x hb
by_cases hx : x ∈ U
· have hh : K (q x,x) ∈ B := by simpa only [G,ite_eq_left hx,one_mul] using hb
have hc := hhits _ (hqb x) x hx hh
rw [hq1 hc] at hh
exact hend x hx hh
· exact houtside x hx (by simpa only [G,ite_eq_right hx] using hb)
end BorsukNine
end OAI
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