Subject: Question for Topology listserv Date: Sun, 24 Aug 2003 13:56:27 -0400 From: "David Hurtubise" I would like to submit the following question to the topology listserv: Does anyone know a necessary (and sufficient?) condition on a Morse function $f:M \rightarrow R$ so that the inclusions in the index filtration are cofibrations? Here's the question in more detail: Let $M$ be a finite dimensional compact smooth Riemannian manifold, and let $f:M \rightarrow R$ be a Morse function. The unstable manifolds $W^u(p)$ of $f$ are embedded open disks in $M$, and the index filtration $F_0 \subseteq F_1 \cdots \subseteq F_m = M$ is defined by $F_k = \cup_{\lambda p \leq k} W^u(p)$ where $\lambda_p$ denotes the index of the critical point $p$. This is, $F_k$ consists of those points in $M$ that lie on gradient flow lines originating from critical points of index less than or equal to $k$. If we assume that $f$ satisfies the Morse-Smale transversality condition, then the index of the critical points decreases along the gradient flow lines, and hence, the endpoint map of the gradient flow from points in $F_{k+1}$ will map into $F_k$. However, the endpoint map of the gradient flow maps to the critical points, and so it doesn't seem to be much help defining a strong deformation retract $R:U \times I \rightarrow U$ where $U$ is some open neighborhood of $F_k$ in $F_{k+1}$. Some authors, i.e. J. Franks, "Morse-Smale Flows and Homotopy Theory", Topology, Vol. 18, p 199-215, have added the condition that the gradient vector field be in "standard form" near the critical points. Basically, this is a generic condition on the metric that says that the coordinate chart coming from the Morse Lemma preserves the Riemannian metric. However, that condition doesn't seem to help much in deciding what happens near the boundaries of the unstable manifolds. I suspect that an answer to this question will involve a certain amount of analysis and/or dynamical systems theory. Any references related to this question would be much appreciated. Thank you, David Hurtubise Department of Mathematics and Statistics Penn State Altoona http://math.aa.psu.edu/ Hurtubise@psu.edu