Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (potentially cellular) map)?


If this setting is to broad, I'm specifically interested in the case, where $M$ is a surface and $X$ is also $2$-dimensional (maybe even restrict it to aspherical surfaces).


Edit: mme provided a counterexample in dimension 3 (homotopy equivalent but not homeomorphic lens spaces), which can probably be generalized to higher dimensions. So only the two-dimensional case remains.



Can we embed a closed manifold into a homotopy equivalent CW complex?
Pick a torus, and add two discs along a meridian and a longitude. You get a 2-complex homotopic to a sphere that does not contain a sphere. This generalises easily to any genus by picking a genus-$g$ surface.


More generally, a finite 2-complex contains finitely many surfaces, and there are some moves (like the Matveev - Piergallini move) that preserve the (simple) homotopy type of the 2-complex, but can modify the surfaces it contains.


