Can we embed a closed manifold into a homotopy equivalent CW complex? 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.
 A: While thinking about it with a friend, we came up with the following two dimensional counter example:
Take the standard knot diagram of the trefoil knot (as a self-intersecting curve in $\mathbb{R}^2$) and let $\bar{X}$ denote the "inner" of this curve i.e. the curve together with the 4 areas bounded by it. Let $X$ denote $\bar{X} \cup_\phi D^2$, where $\phi \colon S^1\to \bar{X}$ follows the trefoil knot. Since $\bar{X}$ is contractible, $X$ is homotopy equivalent to $S^2$. Using cellular homology, one can see that the "fundamental class" of $X$ hits the middle cell of $\bar{X}$ twice, hence there are no injective homotopy equivalences.
By taking "connected sums" of this counterexample with surfaces, one obtains counter examples for all closed surfaces.
A: 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.
