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!

  • $\begingroup$ I don't know specific conditions, but it seems the answer is indeed no, as the product of a real line with en.wikipedia.org/wiki/Dogbone_space is homeomorphic to $\mathbb{R}^4$ $\endgroup$ – Carl Jun 15 '18 at 6:10

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.

  • $\begingroup$ In the MO thread a ref has been added for a pair of (non-manifold) generalized 3-manifolds whose product is a 6-manifold, and this can be done for any $n \geq 6$ with any pair of dimensions $3 \leq k, l$ where $k + l = n$. So unfortunately it is all already known! $\endgroup$ – John Samples Jun 22 '18 at 16:10

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