Maybe some of the following statements are not precise. Please correct them.

Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ satisfies the Morse-Smale condition: for any two critical points $p$ and $q$, the unstable manifold $W^u(p)$ of the negative gradient flow of $f$ of $p$, and the stable manifold $W^s(q)$ of $q$ intersect transversely. Trajectories of this flow that connect $p$ and $q$ will approach to them exponentially.

Now suppose $f$ has a degenerate critical point $p$. Let $q$ still be a non-degenerate critical point. Assume sufficient genericity, $p$ is then of "cubic type", i.e., near $p$, $f$ can be written as $f(x) = f(p) + x_1^3 + \sum_{j\geq 2} \pm x_j^2 +$ higher order terms. Now it seems to me that the unstable set $W^u(p)$ won't be a manifold in general, but a stratified space, with the top stratum= trajectories that approaches to $p$ polynomially(because there is a degenerate direction), and a lower stratum= trajectories that approaches to $p$ exponentially.(Is $W^u(p)$ necessarily a manifold with boundary?)

And now, to ask about the transversality of $W^u(p) \cap W^s(q)$, it should be in a sense of stratified spaces. Is there any original paper that discuss this transversality? I know that Floer has two(maybe more) papers: The unregularized gradient flow of the symplectic action; Morse theory for Lagrangian intersections, which talked about the transversality about non-transverse Lagrangian intersections. Indeed his argument can be carried out for Morse theory. I wonder if there is any more original and more elementary argument about this transversality?

  • $\begingroup$ maybe you can read some papers about Morse homology. $\endgroup$ – yanqing Jun 28 '12 at 4:45
  • 2
    $\begingroup$ @yanqing: which one? $\endgroup$ – Guangbo Xu Jun 28 '12 at 12:05

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