Homotopical Dynamics II: Hopf invariants, smoothings and the Morse complex

从 Morse 函数到余切丛上的 Lefschetz 纤维化

From Morse Functions to Lefschetz Fibrations on Cotangent Bundles

本文证明紧流形上任意 Morse 函数与适配梯度可构造余切丛上同伦唯一的复值辛 Lefschetz 纤维化，并研究其正则纤维的拓扑与辛几何。

Classical Morse theory revisited I -- Backward $λ$-Lemma and homotopy type

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.

Smoothing discrete Morse theory

After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, and conversely, every PL handle vector is also a discrete Morse vector on some PL triangulation. It follows that in dimension up to 7, every discrete Morse vector on some PL triangulation is also a smooth Morse vector; the vice versa is true in all dimensions. This revises and impr…

离散莫尔斯理论与完全图匹配复形的拓扑

Discrete Morse theory and the topology of matching complexes of complete graphs

本文利用离散莫尔斯理论技术研究完全图匹配复形 $M_n$ 的拓扑，证明其具有比同伦连通性更强的几何 $(ν_n-1)$-连通性，并确定 $M_8$ 同伦等价于 132 个 2 维球面的楔和。

带 DG 系数的 Morse 同调

Morse homology with DG coefficients

本文构建带 DG 局部系统系数的 Morse 同调与上同调，证明其等价于 DG Tor/Ext，并建立整数扭曲上闭链、纤维化恢复及对偶等性质。

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincaré--Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that im…