The approximate tubular neighborhood theorem

管状邻域的广义Morse理论

Generalized Morse theory for tubular neighborhoods

该研究针对欧氏空间紧集偏移集的距离函数正则值场景，定义广义Morse函数并构建类Morse理论，用非光滑分析等证明次水平集同伦型在临界值处按每临界点粘一个胞腔的方式改变。

Intersection homology of stratified fibrations and neighborhoods

We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.

Neighborhoods in Stratified Spaces with Two Strata

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h-coborsism extension theorem.

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…