It seems very likely to me that a $n$-dimensional smooth manifold with boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true? Does the manifold need to be compact? What about $n$-dimensional open manifolds?

  • $\begingroup$ It can only true if you require that $M$ is connected or that all components of $M$ have a nonempty boundary. Otherwise it may have a component $S^n$. $\endgroup$ – Paul Frost Dec 27 '18 at 18:25
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    $\begingroup$ @PaulFrost Yes, these obvious assumptions should be added to the question. But then? $\endgroup$ – Yeah Dec 27 '18 at 21:20
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    $\begingroup$ For a discussion in dimension 2, with allusions to higher dimensions, see here. $\endgroup$ – Lee Mosher Dec 27 '18 at 22:45

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