# Homotopy type of smooth manifolds with boundary

It seems very likely to me that every smooth connected $$n$$-dimensional manifold with non-empty boundary has the homotopy type of a $$(n-1)$$-dimensional CW complex. Is that true and how to prove it? (or a counter-example?) Does this manifold need to be compact? What about $$n$$-dimensional open manifolds? Thank you.

• Yes, see here for a combinatorial proof for any noncompact triangulated manifold. When $M$ is compact the proof is easier: the idea is to start with a triangulation and inductively "push in" cells with a free boundary face. – Mike Miller Dec 27 '18 at 21:37
• I wrote out a careful proof of this (based on the answers to my question that Mike Miller linked to) here: www3.nd.edu/~andyp/notes/NoncompactSurfaceFree.pdf – Andy Putman Dec 27 '18 at 23:04

I know of two proofs in the compact case. Let $$M$$ be a compact smooth $$m$$-manifold with boundary $$\partial M$$.

1) Morse theory (Sketch). For this I think we need $$m \ge 4$$. There is a Morse function $$f: M\to \Bbb [0,\infty)$$ with $$f^{-1}(0) = \partial M$$ and $$f^{-1}(1) = \emptyset$$. One can assume $$f$$ is self-indexing. This will give a handlebody structure on $$M$$ relative to $$\partial M$$, i.e., $$M$$ is obtained from attaching handles of index at most $$m$$. By the technique of cancelling $$0$$-handles, one can alter the handle decomposition so that there are no $$0$$-handles. If we turn $$f$$ upside down, we obtain a handle decomposition of $$M$$ with no $$m$$-handles. So $$M$$ is becomes a handlebody whose handles have index $$< m$$. This will imply (by an argument of Milnor) that $$M$$ has the homotopy type of a CW complex of dimension $$\le m-1$$.

2) Wall's theory of finiteness ($$m \ge 4$$). First use that fact that $$M$$ is has the homotopy type of some CW complex (that is well-known and can be proved in different ways). By Poincaré duakity, Wall's condition $$D_{m-1}$$ is satisfied. Conseuqently, $$M$$ has the homotopy type of a CW complex of dimension $$\le m-1$$ by Wall's "Theorem E." Wall's paper can be found here:

C.T.C. Wall, Finiteness Conditions for CW-Complexes. Annals of Mathematics 81 (1965), pp. 56–69