The handle-to-cell comparison across a critical level #

Let f be a continuous function on M and let φ i be finitely many charts with values in ℝⁿ, with pairwise disjoint sources, in which f has the normal form

f = c - Σ_{j < k i} yⱼ² + Σ_{j ≥ k i} yⱼ²,

and whose targets contain the closed ball of radius √ε. Each chart carries the characteristic map χᵢ : (D^{k i}, ∂D^{k i}) → (M_{c+ε}, M_{c-ε}), w ↦ φ i⁻¹(√ε • (w, 0)), of its cell, and its attaching map αᵢ : ∂D^{k i} → M_{c-ε} (OpenPartialHomeomorph.morseCharacteristicMap and OpenPartialHomeomorph.morseAttachingMap), where M_a = {f ≤ a}.

This file forms the space M_{c-ε} ∪_α ⨆ᵢ D^{k i} obtained by attaching one disc along each αᵢ, as a pushout in TopCat (TauCeti.morseCellAttachment), and the comparison map

Φ : M_{c-ε} ∪_α ⨆ᵢ D^{k i} → M_{c+ε}

induced on the pushout by the inclusion of the lower sublevel and the characteristic maps (TauCeti.morseCellComparison). When M is Hausdorff, f is continuous, ε > 0 and the charts have pairwise disjoint sources, Φ is a closed embedding: it is injective because the open discs are carried into the disjoint chart sources and off the lower sublevel, and closed because the lower sublevel is closed and the discs are compact. Its image is the lower sublevel together with the cells {‖φ i x‖² ≤ ε, (φ i x)ⱼ = 0 for j ≥ k i}.

Milnor's cell attachment theorem (TauCeti.exists_sublevel_union_cells_deformation) strong deformation retracts M_{c+ε} onto exactly this image when f is smooth, the charts are smooth and their targets contain the closed ball of radius √(2ε), and the band f⁻¹[c - ε, c + ε] is compact with its critical points in the charts. Transported along Φ, this gives a map g : M_{c+ε} → M_{c-ε} ∪_α ⨆ᵢ D^{k i} with g ∘ Φ = id and a homotopy from Φ ∘ g to the identity fixing the image of Φ, in particular the whole lower sublevel. So Φ is a homotopy equivalence relative to M_{c-ε}; this is the form of the attachment in which a sublevel of a Morse function is built from the previous one by attaching cells. The charts may have any indices, including 0 (a disc attached along the empty sphere) and n, and several critical points on one level are handled simultaneously.

Main definitions #

TauCeti.morseCellAttachment: the space M_{c-ε} ∪_α ⨆ᵢ D^{k i}.
TauCeti.morseCellAttachmentIncl, TauCeti.morseCellAttachmentCell: its inclusion of the lower sublevel and the characteristic maps of its cells.
TauCeti.morseCellAttachmentDesc: the map out of TauCeti.morseCellAttachment induced by compatible maps on the lower sublevel and on the discs.
TauCeti.morseCellAttachmentProdDesc: the map I × (M_{c-ε} ∪_α ⨆ᵢ D^{k i}) → Y induced by compatible maps on I × M_{c-ε} and on the cylinders I × D^{k i}.
TauCeti.morseCellComparison: the comparison map M_{c-ε} ∪_α ⨆ᵢ D^{k i} → M_{c+ε}.
TauCeti.morseCellAttachmentIso: when the attaching maps of two chart families agree strictly up to homeomorphisms h i of the discs, the homeomorphism between their cell attachments which is the identity on the lower sublevel and h i on the cells.

Main results #

TauCeti.isPushout_morseCellAttachment: the pushout square defining the attachment.
TauCeti.isClosedEmbedding_morseCellComparison: the comparison map is a closed embedding.
TauCeti.range_morseCellComparison: its image is the lower sublevel together with the cells.
TauCeti.morseCellAttachmentIso_hom_comp_morseCellComparison: when the characteristic maps are also strictly related, χ₁ i = χ₀ i ∘ h i, the strict reattachment homeomorphism intertwines the comparison maps.
TauCeti.exists_morseCellAttachment_deformation: the comparison map is the inclusion of a strong deformation retract of M_{c+ε}.
TauCeti.IsMorse.exists_morseCellAttachment_deformation: the same for a Morse function at any value c with a compact band (the cells are indexed by the critical points on the level), with the normalized Morse charts and the width ε supplied by TauCeti.IsMorse.exists_admissible_morseCharts.

References #

J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.2 and Remark 3.3.