Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of contractible manifolds (say, open balls), which can be of arbitrary dimension but are homotopy equivalent to a point: however those manifolds are not closed. 



Dimension of two homotopy equivalent manifolds
For a closed, connected topological $n$-manifold, ~~H\_n(X)≠0 (in fact, it's either Z or Z/2Z, depending on orientability)~~ $H\_n(X;\mathbb{Z}/2\mathbb{Z})= \mathbb{Z}/2\mathbb{Z}$ and $H\_m(X)=0$ if $m>n$, hence dimension can be characterised homologically.


Since homology is a homotopy equivalence invariant, so is dimension (for closed, connected topological manifolds).


