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Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.

Is $M $ necessarily homeomorphic to the total space of some vector bundle over a compact manifold?

In fact the only examples I can think up are much more limited, just of the form $M = \Sigma \times \mathbb{R}^n$ where $\Sigma$ is a closed simply connected manifold.

Cross posted on stack exchange https://math.stackexchange.com/questions/4417368/universal-covers-with-one-end.

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I think not. In any dimension $n\geq 4$ there are examples, constructed by Mike Davis, of contractible manifolds $M^n$ that are not homeomorphic to $\mathbb{R}^n$, and yet are the universal cover of a compact manifold $N$. The only way that $M$ could be a vector bundle over a compact closed manifold $X$ is if $X$ were a point and the fiber Euclidean space, which is evidently not true. (I suppose that the base space of the vector bundle would have to be closed, or else $M$ would itself have boundary.)

Davis constructs his manifolds so that they are not simply connected at infinity; this distinguishes them from $\mathbb{R}^n$. At least if $n=4$ this would imply that they are not vector bundles over a non-compact manifold either.

M. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2) 117 (1983), no. 2, 293–324

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  • $\begingroup$ You need some lower bound on $n$ in your second sentence. $\endgroup$
    – Sam Nead
    Commented Apr 6, 2022 at 18:52
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    $\begingroup$ @SamNead Thanks; looks like Will fixed it. I meant $n\geq 4$ as it currently reads. $\endgroup$
    – Danny Ruberman
    Commented Apr 6, 2022 at 19:41
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    $\begingroup$ @WillieWong Thanks; "the" -> "them" should make this linguistically, if not mathematically, clearer. I wasn't intending to summarize Davis' construction, which is fairly complicated. My point was that the property distinguishing Davis's M from $R^n$ shows that they can't be vector bundles. $\endgroup$
    – Danny Ruberman
    Commented Apr 6, 2022 at 21:22
  • $\begingroup$ Thanks for the edit. $\endgroup$
    – Willie Wong
    Commented Apr 7, 2022 at 16:35

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