# Morse theory for manifolds with boundary

I need a reference to some basic facts about Morse theory on manifolds with boundary. Namely, if a critical point lies on the boundary, then the gradient of function might be nonzero and it brings some extra problems.

In fact my manifold is a domain in Euclidean space with smooth boundary, and I can assume that the function is linear. More precisely, I need a reference that could be used in the proof of 4.2.6. here.

Postscript. Let me thank Ryan Budney for the reference to "Morse lemmas for..." by Sergei Vakhrameev; it contains the needed lemmas for manifolds with corners (in particular for manifolds with boundary).

• There are probably more elementary sources, but I suspect whatever you need could be extracted from Goresky-MacPherson’s book “Stratified Morse Theory”. The point is that a manifold with boundary is a stratified space with two strata (the interior, and the boundary). Commented Aug 11, 2023 at 15:41
• Have a look at the local model $f(x_1,\dotsc, x_n)=\pm x_1+\sum_{j=2}^n \pm x_j^2$ in the halfspace $\{x_1\geq 0\}$ Commented Aug 11, 2023 at 17:59
• There's a highly illuminating account in Kronheimer-Mrowka, "Monopoles and 3-Manifolds", introductory chapter. Commented Aug 11, 2023 at 20:11
• The main issue with manifolds with boundary is you get some additional cells where the gradient of the function (restricted to the boundary) is zero. You only get cells if the gradient of the original function is pointing "out" of the manifold. Well, this is for the upwards flow, i.e. flow with gradient. If you take the downwards flow then you get cells when the gradient is pointing in. You can express Alexander duality quite nicely via this out/in duality. Commented Aug 11, 2023 at 21:24
• I wrote a paper using Morse theory on manifolds with cubical corners, so one step up from manifolds with boundary. As references I used Goresky-MacPherson, also Handron's "Generalized billiard paths and Morse theory on manifolds with corners" as well as Vakhrameev "Morse lemmas for smooth functions on manifolds with corners". I forget which reference had which results, but I think between Handron and Vakhrameev you should find everything you need. Commented Aug 12, 2023 at 6:42

• This statement seems obvious (but maybe you mean another one?): it says that given a critical value $f(c)$ of $f\vert\partial M$ where $df\cdot n$ (the exterior-pointing normal) is positive, the spaces $M_{c-\epsilon}=\{f\le c-\epsilon\}$ and $M_{c+\epsilon}$ have the same homotopy type. Actually the second retracts by deformation on the first by following the lines of a properly defined gradient vector field, no? Commented Aug 12, 2023 at 11:47