[Bing gave](https://www.jstor.org/stable/1970322) a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is best possible? In other words, if $X \times Y =Z$ where $X$ is a manifold and $Z$ is a 3-manifold, then is $Y$ a manifold?


In Bing's example, $Z$ is not compact. Is there a compact example in dimension $4$? In the example, $X$ is the real line, so one can also ask if it possible to get a four dimensional example where $X$ is a surface, or where $X$ is compact.



Lowest Dimension for Counterexample in Topological Manifold Factorization
As asked Dusan Repovs (who is an expert in the theory of topological manifolds), and he sent me the following answer:


This is indeed best possible result, since whenever a product of two spaces is a topological manifold, both factors must be generalized manifolds - which in dimensions below 3 are topological manifolds.


Ref.: A.Cavicchioli, F.Hegenbarth and D.Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions, EMS Series of Lectures in Mathematics, European Mathematical Society, Zurich, 2016. 


To the second question: there are also compact examples (in dimensions >3): e.g. take the product of the 3-sphere $S^3$ modulo the Fox-Artin wild arc $A$ and $S^1$. This product is homeomorphic to $S^3\times S^1$.


Ref.: R.J.Daverman, Decompositions of Manifolds, Academic Press, Orlando, 1986.




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**Added in Edit:** Answering a comment of John Samples, Dusan Repovs pointed out that the Chapter 29 of Daverman's book contains the following fact: for any $n,m>2$ there are a generalized $n$-manifold $X$ and a generalized $m$-manifold $Y$ which are not topological manifolds, but their product $X\times Y$ is a topological manifold.


