管状邻域的广义Morse理论

The approximate tubular neighborhood theorem

Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by mapping cylinder neighborhoods. Thus, this is the best possible topological tubular neighborhood theorem in the stratified setting.

局部Morse上同调中的相对杯长

The relative cup-length in local Morse cohomology

该研究证明局部Morse上同调是孤立邻域上同调的模，定义了相对杯长作为非必为Morse函数临界点数量的下界，并通过实例表明该下界强于绝对杯长给出的下界。

Generalizing the hyperbolic collar lemma

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number $k$ on a hyperbolic surface. This lower bound tends to infinity with $k$. Our second generalization is to totally geodesic hypersurfaces of hyperbolic manifolds. Namely, we construct a tubular neighborhood funct…

Twisted Novikov homology and circle-valued Morse theory for knots and links

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). Novikov homology provides lower bounds for MN(L). In the present paper we introduce the notion of twisted Novikov homology, which allows to obtain better lower bounds for MN(L) than the usual Novikov homology. Our twisted Novikov homology is a module over the Novikov ring Z((t)) bu…

无奇点的圈值Morse函数扩展

Non-singular extensions of circle-valued Morse functions

研究闭合定向曲面上的圈值Morse函数非奇点扩展条件，给出在 collar 邻域存在子浸入时的必要充分条件。

Stratified Morse Theory with Tangential Conditions

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the local unstable set is a manifold and the local stable set is itself an abstract stratified space. We also give a normal form for the gradient dynamics in the neighborhood of critical points. For a stratified Morse pair which satisfies the generic Morse-Smale condition we can build t…