As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds.

If we have a manifold $M$, such that $M$ is homeomorphic to $X \times Y$, then must $X$ and $Y$ be manifolds? The converse ($X,Y$ manifolds implies $X \times Y$ is a manifold) is certainly true. I'd like to think it is true, but I have seen enough strange topological behaviour to suggest this may not be true.

For this question take 'manifold' to mean a second countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^n$, for some finite $n$.

  • $\begingroup$ Thanks for following up on this! My intuition told me it was false, but maybe I'm just a cynic when it comes to spaces that aren't at least CW complexes. Or does that make me an optimist? I guess it all depends on your perspective :o) $\endgroup$ – Aaron Mazel-Gee Nov 4 '11 at 5:59
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    $\begingroup$ Juan, @Aaron: In case you missed it: there's a new related thread on MO. $\endgroup$ – t.b. Dec 17 '11 at 3:13

Disclaimer: I'm by no means knowledgeable in this field and I haven't read the papers or books I mention below. I found these by digging in the literature and hope these pointers are useful.

The answer to your question is no.

  1. The first example was given by R.H. Bing, The Cartesian Product of a Certain Nonmanifold and a Line is $E^4$, Ann. of Math. (2) 70 (1959) 399–412. MR107228.

    Bing describes a topological space $B$ — in fact a quotient space of $\mathbb{R}^3$, sometimes called the Dogbone space — which is not a manifold and has the property that $B \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^4$.

  2. Modifying this construction and relying heavily on work of Andrews and Curtis, K.W. Kwun, Product of Euclidean Spaces Modulo an Arc, Ann. of Math. (2) 79 (1964) 104–108, MR159312, produced product decompositions of $\mathbb{R}^n \cong X \times Y$ for $n \geq 6$ where neither $X$ nor $Y$ is a manifold.

  3. Here are two freely available papers by A.J. Boals:

  4. Quoting C.D. Bass, Some products of topological spaces wich are manifolds, Proc. Amer. Math. Soc. 81 (1981), 641–646, MR601746, Corollary 3 on page 645 “gives abundant examples of factorizations of certain manifolds into nonmanifold factors.”

  5. A textbook covering these and many more topics:

    Daverman, Robert J., Decompositions of manifolds, Pure and Applied Mathematics, 124, Academic Press, 1986, MR872468. (Reprinted by the AMS, 2007).

Here are two related MO-threads:

Added: A further MO-thread was posted and answered a couple of hours ago:

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    $\begingroup$ Thanks Theo! Nice detective work...I thought about searching through all of Bing's paper's. I had a feeling if it was not true he would have provided a counter example $\endgroup$ – Juan S Nov 2 '11 at 21:44
  • $\begingroup$ @Juan (Qwirk): you're welcome. I wanted to look up this stuff for quite a while, so now I had the opportunity... Interesting stuff, but the details seem somewhat messy. $\endgroup$ – t.b. Nov 2 '11 at 22:10
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    $\begingroup$ Great answer t.b.! You write "I'm by no means knowledgeable in this field" What would it be if you were:-) $\endgroup$ – Georges Elencwajg Nov 2 '11 at 23:22
  • $\begingroup$ Dear @Georges: Thank you, but yes, I meant it. I know little about geometric topology and the question at hand. I knew the Dogbone space and the two MO-questions, but not more. $\endgroup$ – t.b. Nov 2 '11 at 23:24

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