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Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times [0,\infty)$. I know that if $M$ is compact then it is forced to be $\Sigma_2\times I$, is a similar result with $\Sigma_2\times [0,1)$ true?

user333828
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  • You have virtually zero control over the topology of noncompact 3-manifolds (there are already uncountably many irreducible contractible 3-manifolds). So it is extremely unlikely your proposed conjecture is true, though I've made no effort to write down an explicit counterexample. If you're feeling motivated, try to modify a Whitehead manifold to make your question false. –  Apr 22 '16 at 21:15
  • I think you need stronger assumptions on your manifold in order to deduce this. But I also believe that Berni Waterman's thesis is at least partly about this topic, I'll ask him whether he could shed some light on the assumptions one would need to deduce this, which was also your question. – Daniel Valenzuela Apr 23 '16 at 07:06

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While this isn't a duplicate of the question, the question is answered by the user studiosus here: A 3-manifold with fundamental group isomorphic to a surface group.

I'll mention that the existence of a compact 3-manifold $S\subset M$ (with boundary) with the inclusion a homotopy equivalence is a consequence of Scott's Core theorem.

G.P. Scott ,"Compact submanifolds of 3-manifolds", J. London Math. Soc. (2) 7 (1973) 246-250

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As a counter example was already presented, I'll focus on the last part of your question, and conclude with a (a priori large) class of manifolds which will not be able to serve you with counter examples.

Claim: The result holds if and only if $M$ is tame.

One direction is immediate. For the other direction, let $M$ be tame and hence properly embeds $M\hookrightarrow M'$, where $M'$ is compact and the image of $M$ contains the interior of $M'$. By compactness we have $M'=\Sigma_g\times I$ and hence $M=\Sigma_g\times [0,\infty)$.

In particular, as the tameness conjecture was proven by Agol a little more then 10 year ago, we know that we have:

Theorem: Let $M$ be any hyperbolic 3-manifold with $\partial M =\Sigma_g$ and $\pi_1M=\pi_1\Sigma_g$. Then $M\cong \Sigma_g\times [0,\infty)$.

Daniel Valenzuela
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  • This is all correct except the correct reference is not Agol but: F. Bonahon. Bouts des vari´et´es hyperboliques de dimension 3. Ann. of Math. (2), 124(1): 71–158, 1986. – Moishe Kohan Jun 15 '16 at 16:37