Factorization of Products of Manifolds This is a fairly general question but I couldn't find a source that deals with this problem in the sense I was wondering.
Suppose that we are given two topological spaces $X, Y$ such that $X \times Y = Z$ is a connected manifold of dimension $n > 1$.  If $X$ is a manifold, must $Y$ be a manifold?  My guess is no, but what conditions on either $Y$ or the pair $X, Y$ are sufficient?
Cases of particular interest to me are $Z$ is a 3-manifold or $Z$ is compact.
Thanks in advance!
 A: As per https://mathoverflow.net/questions/302990/lowest-dimension-for-counterexample-in-topological-manifold-factorization
1) There are no examples for dim$(Z) = 3$ or less, since any topological factor of a manifold is a generalized (homology) manifold, and in dimension less than $3$ this concept coincides with manifold.
2) There is a compact example in every dimension $n \geq 4$ of a manifold $Z = X \times Y$ where $X$ is a manifold but $Y$ is not.  In fact, $Z$ can be chosen to be $\mathbb{S}^n$.
For references, see the link.  The compact examples involve complements of wild arcs.
There are some nice, immediate corollaries of (1) using a bit of dimension theory, which will apply in the case of homology manifolds.
3) If dim$(Z) = 4$ such that $X$ is a surface and $Z$ is a manifold, then $Y$ is a manifold.
4) If dim$(Z) \leq 5$ then at least one of the factors is a manifold.
This leaves open, then, the possibility of two non-manifolds producing a manifold in dimension $6$.  The constructions of the compact cases are done with spaces of dimension $1$ and $n-1$.  This leaves open another question regarding products of spaces with identical dimensions for even $n$, and similar questions that may give geometric insight into the different behavior of 'medium-dimensional' spaces compared to 'very high-dimensional' ones.
Edit: see the comment below.
