It is known that self homotopy equivalence of closed surfaces is homotopic to a self homeomorphism. I wonder whether this statement holds for all closed manifolds. Note that no smoothness of the self homeomorphism is required.
I suspect this statement would fail in a low dimension. A potential conterexample is the map $X \times Y \xrightarrow{f \times g} Y \times X \xrightarrow{\sigma} X \times Y$, where $f: X \rightarrow Y$ and $g: Y \rightarrow X$ are homotopy equivalences between two closed manifolds (e.g. $L(7,1)$ and $L(7,2)$), and $\sigma: Y \times X \rightarrow X \times Y$ is the exchange map. Is this example valid? And what is the greatest dimension such that this statement still holds true?
There are self-homotopy equivalences of some 3-dimensional lens spaces (specifically, $L(12,1)$) which are not homotopic to self-homeomorphisms:
Darryl McCullough. "Mappings of reducible 3-manifolds." Banach Center Publications 18 (1986), p. 61-76.