# Homotopy type of smooth manifold with boundary

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?

• 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 $S^n$. – Paul Frost Dec 27 '18 at 18:25
• @PaulFrost Yes, these obvious assumptions should be added to the question. But then? – Yeah Dec 27 '18 at 21:20
• For a discussion in dimension 2, with allusions to higher dimensions, see here. – Lee Mosher Dec 27 '18 at 22:45