Response to question posted earlier this week...........DMD ______________________________________________________ Subject: Re: question abt Morse fcns Date: Mon, 25 Aug 2003 11:59:20 -0400 (EDT) From: Walter Neumann I seem to remember a long article by Gudrun Kalmbach (late 60's or early 70's) that probably gives an answer to this. I think it was a chapter in a collection (a conference proceedings?); it is neither in Math Reviews nor Zentralblatt. --walter neumann On Mon, 25 Aug 2003, Don Davis wrote: > 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 > >