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In this question, I asked for interesting / non-trivial examples of smooth connected closed manifolds that happen to be direct products or involve direct products, especially orientable manifolds.

In all answers there, one of the factors is a sphere $S^n$, including $S^1$. Now my question is whether there are such examples where both factors are less trivial -- for example, have non-trivial (and not $\mathbb Z$) fundamental group.

Typically an "important" manifold would have a proper name or standard notation or would be involved in some interesting theorems or examples or have other applications.

Motivation: I work on a paper involving the fundamental group of direct product, and I need to explain to the reviewers why the topic is important and where it can be applied.

Interesting examples would probably have dimension at least 5. In dimension 4, these are only $M^2_{g_1}\times M^2_{g_2}$ (surfaces of genus $g_i$).

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Alexander Gelbukh
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  • Isn't the fundamental group of a direct product the direct product of fundamental groups? – Theo Johnson-Freyd Mar 07 '15 at 02:07
  • @TheoJohnson-Freyd Yes it is. – Alexander Gelbukh Mar 07 '15 at 02:08
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    So can I ask what else you say about it in your paper? I'm having a hard time seeing any interesting questions there are to answer, but perhaps I'm missing something important. In low dimensions there surely are things to say, since 2- and 3-manifold groups cannot be completely general. But in high dimensions, I think that every finitely-presented group is the fundamental group of a 4-manifold. Or do you say something in the converse? Something like using factorizations of the fundamental group to factor the manifold? – Theo Johnson-Freyd Mar 07 '15 at 02:15
  • Well, it's a bit off-topic to explain my paper here. I just need a non-trivial example, and I thought it would be useful for other readers, too. – Alexander Gelbukh Mar 07 '15 at 02:25
  • Fare enough. I was just being nosy. – Theo Johnson-Freyd Mar 08 '15 at 15:30

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Something I probably should have mentioned in the previous answer: Smale's H-cobordism theorem.

Suppose $M_1$ and $M_2$ are closed simply connected manifolds of dimension at least $5$. Suppose $W$ is a smooth manifold with boundary $\partial W = M_1 \coprod M_2$ and for which the natural inclusions $i:M_i\rightarrow W$ are both homotopy equivalences. Then, $M_1$ is diffeomorphic to $M_2$ and $W$ is diffeomorphic to a product $W \cong M_1\times [0,1]$.

(Such a $W$ is a called an $H$-cobordism between $M_1$ and $M_2$, hence the name of the theorem).

My understanding is that if the $M_i$ have dimension $4$, or are not simply connected, then the conclusion is false. So, in this sense, the fact that the manifold $W$ is a product is quite surprising.

This theorem has, as one of its many consequences, the Poincare conjecture in dimension $6$ and above. (Sketch: Start with a homotopy sphere $\Sigma$ of dimension $n\geq 6$. Delete two non-overlapping charts to get a manifold $\Sigma \setminus U_1\cup U_2$ with boundary $S^{n-1} \coprod S^{n-1}$. Prove that this manifold with boundary is an $H$-cobordism, so $\Sigma\setminus U_1\cup U_2 \cong S^{n-1}\times [0,1]$. Now, glue back in the charts to see that $\Sigma = D^n \cup_f D^n$ for some gluing diffeomorphism $f$. Then the Alexander trick shows $\Sigma$ is homeomorphic to $S^n$.)

Jason DeVito - on hiatus
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  • Thank you! I don't quite get the answer, could you please clarify? The question was about examples of closed manifolds with non-trivial fundamental group. In your answer, $W$ seems to be not closed, $M_i$ seem to have trivial fundamental group, and I don't quite see an example. You seem to imply that there should not exist interesting examples in dimension 4 (right, all such examples are $M^2_{g_1}\times M^2_{g_2}$). – Alexander Gelbukh Mar 07 '15 at 18:02
  • I see, I thought you were just trying to one one of the factors to not be a sphere. Everything here is simply connected, sorry. I can delete, if you want – Jason DeVito - on hiatus Mar 07 '15 at 21:07
  • Please don't delete, it's great to share your knowledge (upvoting). This info can be useful for the readers. Just it does not answer this specific question. – Alexander Gelbukh Mar 07 '15 at 21:22
  • Fair enough (and thanks for the upvote). In practice, I almost exclusively think about simply connected manifolds. I'll try to think of something a bit more interesting, but nothing is currently coming to mind :-(. – Jason DeVito - on hiatus Mar 07 '15 at 21:33
  • Same here. OK, waiting. If there is nothing, then, maybe, my paper is on a topic that is not really important :-( – Alexander Gelbukh Mar 07 '15 at 21:36