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I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S_g$ must itself already be homemorphic to $S_g \times \mathbb R$. There seems to be a very deep theorem behind it, however, I couldn't find any reference which would bring this up. Does anybody know a good reference for this ?

Edit: The original question has been answered well. I wonder in how far the assumptions have to be strengthened in order for this to be true. The Tameness Theorem shows that $X = S_g \times \mathbb R$ whenever $X$ additionally admits a complete hyperbolic metric. However, for my purposes, it would be nice to have purely topological assumptions. I have the following suggestions:

$X$ is as above, but additionally we require $X$ to have two ends, such that for each end, there exists a sequence of end-neighborhoods $U_1 \supset U_2 \supset U_3 \supset ....$ with $\bigcap cl(U_i) = \emptyset$ and the following properties:

  • $U_1$ is homotopy equivalent to $S_g$
  • the natural map $i_*: \pi_1(U_{n+1}) \to \pi_1(U_n)$ is an isomorphism for all $n$.

Intuitively, this will prevent $X$ from "going crazy" at one end. Is this enough now ?

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  • $\begingroup$ You'd better require that the 3-manifold itself is orientable. $\endgroup$
    – Allen Knutson
    Commented Nov 11, 2015 at 10:45
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    $\begingroup$ Let $T=S/u$ be a closed surface, where $S$ is an orientable closed surface and $u$ a fixed-point free involution. Then the quotient $X$ of $S\times\mathbf{R}$ by the involution $v(x,t)=(u(x),-t)$ is an open 3-fold, and is homotopy equivalent to $T$, and has only one end, so is not homeomorphic to $T\times\mathbf{R}$. Note that if $T$ is orientable, then $X$ is non-orientable. $\endgroup$
    – YCor
    Commented Nov 11, 2015 at 10:58
  • $\begingroup$ Yes, I should require orientability. Thanks for pointing that out. $\endgroup$
    – H1ghfiv3
    Commented Nov 11, 2015 at 11:42
  • $\begingroup$ If you assume that your manifold is the interior of a compact manifold $M$, then I think $M$ is an $I$-bundle, and in most cases the proof is in Chapter 10 of Hampel's "3-manifolds" (combined with nonexistence of fake 3-cells due to Perelman). This is enough to handle the case when $g\ge 1$. As Igor Rivin points out there are nontame 3-manifolds with surface fundamental group, so the tameness assumption is necessary. $\endgroup$
    – Igor Belegradek
    Commented Nov 11, 2015 at 14:05
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    $\begingroup$ A 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. See L. S. Husch and T. M. Price, Finding a boundary for a 3 -manifold, Ann. of Math. (2) 91 (1970), 223–235 for conditions that imply tameness. It may be that your condition is sufficient. $\endgroup$
    – Igor Belegradek
    Commented Nov 11, 2015 at 15:21

2 Answers 2

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The answer (by studiosus) to this question on MSE gives references AND a counterexample!

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  • $\begingroup$ It should be true for hyperbolic manifolds because the Calegari-Gabai-Agol theorem (aka tameness conjecture) then reduces the problem to compact (irreducible and orientable) 3-manifolds with boundary, which must indeed be homeomorphic to SxI. $\endgroup$
    – ThiKu
    Commented Nov 11, 2015 at 11:57
  • $\begingroup$ So one must additionally require that the manifold in question is the interior of a compact manifold-with-boundary ? $\endgroup$
    – H1ghfiv3
    Commented Nov 11, 2015 at 12:00
  • $\begingroup$ Yes. In the MSE-answer they also required irreducibility. But this was just because the question there didn't assume homotopy equivalence but only an isomorphism of fundamental groups. $\endgroup$
    – ThiKu
    Commented Nov 11, 2015 at 12:28
  • $\begingroup$ I have edited my answer and suggested an additonal assumption. You might like to check it out. $\endgroup$
    – H1ghfiv3
    Commented Nov 11, 2015 at 12:31
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This paper gives some other very nice counter-examples. As I recall, for open manifolds you really need to think about proper homotopies, not just homotopies.

MR1033220 (91b:57021) Reviewed Scott, Peter(1-MI); Tucker, Thomas(1-COLG) Some examples of exotic noncompact 3-manifolds. Quart. J. Math. Oxford Ser. (2) 40 (1989), no. 160, 481–499. 57N10 (57M10)

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