Examples of important manifolds that are direct products of non-trivial manifolds

Question

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 $g_i$).

Answer 1

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 $S^n$.)