Attaching maps of Morse cells in homology under a change of chart

Let ψ₀ and ψ₁ be two normal-form charts of index m + 1 at a critical point of a function f, with characteristic maps χ₀, χ₁ : (D^{m+1}, ∂D^{m+1}) → (M_{c+ε}, M_{c-ε}) and attaching maps α₀, α₁ : ∂D^{m+1} → M_{c-ε}, where M_a = {f ≤ a}. In Milnor's comparison of the two charts, the characteristic maps are related by a homotopy of maps of pairs from χ₁ to χ₀ ∘ g for an orthogonal map g of ℝ^{m+1}, the polar factor of the negative block of the derivative of the transition. On the boundary sphere such a homotopy is a homotopy from α₁ to α₀ ∘ g (TauCeti.homotopic_morseAttachingMap_diskBoundaryMap_comp).

The boundary map of g has degree sign (det g) = ±1: it acts on the reduced homology H_redₘ(∂D^{m+1}) by that sign (TauCeti.reducedSingularHomologyFunctor_map_diskBoundaryMap). So the maps induced by the two attaching maps on reduced homology, with coefficients in any abelian category with coproducts, differ by the sign of det g (TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_eq_sign_det_smul). In particular, an orientation-reversing change of chart, det g = -1, changes the sign of the attaching map in homology (TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_of_det_eq_neg_one), and an orientation-preserving one does not. This holds in every index m + 1 ≥ 1. For index 1 the sphere ∂D¹ consists of two points, g = -1, and the attaching maps are related by the swap of the two points (TauCeti.homotopic_morseAttachingMap_diskBoundaryNeg_comp_of_det_eq_neg_one); on H_red₀(∂D¹) the swap acts by -1.

Main results

• TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_eq_sign_det_smul: the attaching maps of two charts related by g agree on reduced homology up to the sign of det g.
• TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_of_det_eq_neg_one: if det g = -1, the attaching maps induce opposite maps on reduced homology.

References

• J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.2 and Remark 3.3.
• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002, Section 2.2, properties of the degree.

theorem TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_eq_sign_det_smul {C : Type w} [CategoryTheory.Category.{v, w} C] [CategoryTheory.Limits.HasCoproducts C] [CategoryTheory.Abelian C] (R : C) {M : Type u} [TopologicalSpace M] {n m : ℕ} {f : M → ℝ} {c ε : ℝ} {ψ₀ ψ₁ : OpenPartialHomeomorph M (EuclideanSpace ℝ (Fin n))} (hk : m + 1 ≤ n) (hε : 0 ≤ ε) (hball₀ : Metric.closedBall 0 √ε ⊆ ψ₀.target) (hball₁ : Metric.closedBall 0 √ε ⊆ ψ₁.target) (hf₀ : ∀ y ∈ ψ₀.source, f y = c + ∑ j : Fin n, (if ↑j < m + 1 then -1 else 1) * (↑ψ₀ y).ofLp j ^ 2) (hf₁ : ∀ y ∈ ψ₁.source, f y = c + ∑ j : Fin n, (if ↑j < m + 1 then -1 else 1) * (↑ψ₁ y).ofLp j ^ 2) {g : EuclideanSpace ℝ (Fin (m + 1)) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin (m + 1))} (K : (TopCat.Hom.hom (ψ₁.morseCharacteristicMap hk hε hball₁ hf₁)).Homotopy (TopCat.Hom.hom (CategoryTheory.CategoryStruct.comp g.diskMap (ψ₀.morseCharacteristicMap hk hε hball₀ hf₀)))) (hK : ∀ (t : ↑unitInterval) (s : ↑(TopCat.diskBoundary (m + 1))), f ↑(K (t, (CategoryTheory.ConcreteCategory.hom (TopCat.diskBoundaryInclusion (m + 1))) s)) ≤ c - ε) : (reducedSingularHomologyFunctor R m).map (ψ₁.morseAttachingMap hk hε hball₁ hf₁) = ↑(SignType.sign (LinearMap.det ↑g.toLinearEquiv)) • (reducedSingularHomologyFunctor R m).map (ψ₀.morseAttachingMap hk hε hball₀ hf₀)

A change of chart multiplies the attaching map in homology by the sign of its orientation. Let χ₀ and χ₁ be the characteristic maps of the cells of two normal-form charts of index m + 1, and suppose χ₁ is homotopic, as a map of pairs into (M_{c+ε}, M_{c-ε}), to χ₀ ∘ g for a linear isometry g of ℝ^{m+1}. Then the attaching maps α₀, α₁ : ∂D^{m+1} → M_{c-ε} satisfy α₁_* = sign (det g) • α₀_* on the reduced homology H_redₘ.

theorem TauCeti.reducedSingularHomologyFunctor_map_morseAttachingMap_of_det_eq_neg_one {C : Type w} [CategoryTheory.Category.{v, w} C] [CategoryTheory.Limits.HasCoproducts C] [CategoryTheory.Abelian C] (R : C) {M : Type u} [TopologicalSpace M] {n m : ℕ} {f : M → ℝ} {c ε : ℝ} {ψ₀ ψ₁ : OpenPartialHomeomorph M (EuclideanSpace ℝ (Fin n))} (hk : m + 1 ≤ n) (hε : 0 ≤ ε) (hball₀ : Metric.closedBall 0 √ε ⊆ ψ₀.target) (hball₁ : Metric.closedBall 0 √ε ⊆ ψ₁.target) (hf₀ : ∀ y ∈ ψ₀.source, f y = c + ∑ j : Fin n, (if ↑j < m + 1 then -1 else 1) * (↑ψ₀ y).ofLp j ^ 2) (hf₁ : ∀ y ∈ ψ₁.source, f y = c + ∑ j : Fin n, (if ↑j < m + 1 then -1 else 1) * (↑ψ₁ y).ofLp j ^ 2) {g : EuclideanSpace ℝ (Fin (m + 1)) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin (m + 1))} (hg : LinearMap.det ↑g.toLinearEquiv = -1) (K : (TopCat.Hom.hom (ψ₁.morseCharacteristicMap hk hε hball₁ hf₁)).Homotopy (TopCat.Hom.hom (CategoryTheory.CategoryStruct.comp g.diskMap (ψ₀.morseCharacteristicMap hk hε hball₀ hf₀)))) (hK : ∀ (t : ↑unitInterval) (s : ↑(TopCat.diskBoundary (m + 1))), f ↑(K (t, (CategoryTheory.ConcreteCategory.hom (TopCat.diskBoundaryInclusion (m + 1))) s)) ≤ c - ε) : (reducedSingularHomologyFunctor R m).map (ψ₁.morseAttachingMap hk hε hball₁ hf₁) = -(reducedSingularHomologyFunctor R m).map (ψ₀.morseAttachingMap hk hε hball₀ hf₀)

An orientation-reversing change of chart negates the attaching map in homology. Let χ₀ and χ₁ be the characteristic maps of the cells of two normal-form charts of index m + 1, and suppose χ₁ is homotopic, as a map of pairs into (M_{c+ε}, M_{c-ε}), to χ₀ ∘ g for a linear isometry g of ℝ^{m+1} with det g = -1. Then the attaching maps α₀, α₁ : ∂D^{m+1} → M_{c-ε} induce opposite maps α₁_* = -α₀_* on the reduced homology H_redₘ: the boundary map of g has degree -1.