Convexity of Morse Stratifications and Gradient Spines of 3-Manifolds
| dc.creator | Katz, Gabriel | |
| dc.date | 2006-10-31 | |
| dc.date | 2008-02-19 | |
| dc.date.accessioned | 2026-07-07T09:21:36Z | |
| dc.date.available | 2026-07-07T09:21:36Z | |
| dc.description | We notice that a generic nonsingular gradient field $v = \nabla f$ on a compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$ of $X$. We study the transformations of $K(f, v)$ that are induced by deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$ with counting the \emph{double-tangent} trajectories of the $v$-flow, i.e. the trajectories that are tangent to the boundary $\d X$ at a pair of distinct points. Let $gc(X)$ be the minimum number of such trajectories, minimum being taken over all nonsingular $v$'s. We call $gc(X)$ the \emph{gradient complexity} of $X$. Next, we prove that there are only finitely many $X$ of bounded gradient complexity, provided that $X$ is irreducible and boundary irreducible with no essential annuli. In particular, there exists only finitely many hyperbolic manifolds $X$ with bounded $gc(X)$. For such $X$, their normalized hyperbolic volume gives an upper bound of $gc(X)$. If an irreducible and boundary irreducible $X$ with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then $X$ is a standard ball. All these and many other results of the paper rely on a careful study of the stratified geometry of $\d X$ relative to the $v$-flow. It is characterized by failure of $\d X$ to be \emph{convex} with respect to a generic flow $v$. It turns out, that convexity or its lack have profound influence on the topology of $X$. This interplay between intrinsic concavity of $\d X$ with respect to any gradient-like flow and complexity of $X$ is in the focus of the paper. | |
| dc.description | 60 pages, 35 figures, pdfsync.sty, graphix.sty | |
| dc.identifier | https://arxiv.org/abs/math/0611005 | |
| dc.identifier | http://arxiv.org/abs/math/0611005 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155078 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57M27, 57M20 | |
| dc.title | Convexity of Morse Stratifications and Gradient Spines of 3-Manifolds | |
| dc.type | text |