Let $M$ be a compact $3$-manifold with nonempty boundary. If $\pi_1(M)=\mathbb Z$, can we prove that $M$ is homeomorphic to $S^1 \times D^2$?
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3$\begingroup$ Theorem 5.3 in Hempel's book should answer this completely. $\endgroup$– Steve DCommented Jan 25, 2021 at 16:07
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$\begingroup$ You can also look at its universal covering space. With the right kinds of extra hypotheses (see Sam Nead's post) it will look like a very long sausage and that will imply your result. Edit: Some clarifications are added below. The argument in Hatcher's notes use homology. Here is another approach which doesn't involve homology. Dehn's lemma is still needed. Let $M$ be your 3 manifold and let $\tilde M$ be its universal cover. If $\tilde M$ looks like a very long sausage i.e. its boundary is a cylinder $\partial\tilde M = S^1\times \mathbb R$, then you've got the desired result. Now you wi $\endgroup$– NWMTCommented Jan 26, 2021 at 15:04
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2 Answers
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No. For example, take a copy of $S^1 \times S^2$ and remove the interior of a closed, nicely embedded, three-ball.
You will need to add the hypothesis of irreducibility (to rule out "punctures" as in my example immediately above) and the hypothesis of orientability (to rule out the solid Klein bottle). These hypotheses, plus the disk theorem, gives the desired result.
See Proposition 3.4 of Hatcher's three-manifold notes, Exercise 5.3 in Hempel's book, or Exercise I.32 in Jaco's book.
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See for instance Kawauchi's article A classification of compact 3-manifolds with infinite cyclic fundamental groups. However, by looking at the review on mathscinet:
The author provides a somewhat lengthy proof of the well-known fact that compact 3-manifolds with infinite cyclic fundamental group can be written as a connected sum of a simply connected 3-manifold with a bundle having a circle base and fiber a simply connected 2-manifold.
there may be a better reference.
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$\begingroup$ Is the proof really lengthy? If so, is it just detailed, or not optimal? $\endgroup$– YCorCommented Jan 25, 2021 at 15:39
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$\begingroup$ I don't really know actually. I am not sufficiently familiar with 3-manifolds to have a relevent opinion. $\endgroup$ Commented Jan 25, 2021 at 21:30
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1$\begingroup$ The proof in this paper seems far more complicated than the one in Hempel's book which is just a simple application of the loop theorem. This paper also uses the loop theorem and depends as well on something the author calls Partial Poincaré Duality. This doesn't seem to be a well known concept and the only reference given for it is the author's master's thesis. A mathscinet search turned up a published version in the 1975 Oxford Quarterly. It involves infinite cyclic coverings of manifolds and generalizes a paper of Milnor on infinite cyclic coverings. The statement is not simple. $\endgroup$ Commented Feb 20, 2021 at 22:02
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$\begingroup$ @AllenHatcher: Thanks for the details. $\endgroup$ Commented Feb 21, 2021 at 8:00
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$\begingroup$ @AllenHatcher I posted a Meta question on MSE about merging your new spurious account into the existing one there. Asaf answered that you should follow the directions at math.stackexchange.com/help/merging-accounts in reference to math.stackexchange.com/users/95137/allen-hatcher and new math.stackexchange.com/users/890442/user890442 $\endgroup$ Commented Feb 22, 2021 at 21:31