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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.

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    $\begingroup$ 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. $\endgroup$ – Mike Miller Dec 27 '18 at 21:37
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    $\begingroup$ 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 $\endgroup$ – Andy Putman Dec 27 '18 at 23:04
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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

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