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