# Smooth structures on $M\times S^1$

Let $M$ be a smooth $n$-manifold. Here are three constructions which produce manifolds which are homeomorphic to $M\times S^1$, but might not be diffeomorphic to it:

1. Take $M'\times S^1$, where $M'$ is homeomorphic to, but not diffeomorphic to $M$.

2. Take a connected sum of $M\times S^1$ with an exotic $n+1$-sphere.

3. Take the mapping torus of a self-diffeomorphism of $M$ which is homotopic, but not (differentiably) isotopic to the identity.

I would like to know two things:

Question A: when do these constructions actually produce new smooth structures?

Question B: which manifolds have the property that every smooth structure on them can be obtained by operations 1-3 above?

I would be interested both in theoretical answers with references to the literature, and explicit answers for specific manifolds, such as $S^6$, or other large-but-not-too-large dimensional spheres.

Remark 1: question A for construction 2 (connected sums) was essentially already asked here. This problem is apparently known among experts as "computing inertia groups". So I think part of my question is: are the inertia groups of $S^n\times S^1$ known?

Remark 2: question A for construction 3 is closely related to this question The difference is that I am asking about the diffeomorphism type of the total space, not fibered equivalence.

I assume that you are interested in this question in high dimensions, which should be 6 (or possibly 5) for $M \times S^1$. Construction 2 is actually a special case of construction 3, for if you take $(M \times S^1) \# \Sigma$ (the last being a homotopy sphere) then you can split along a copy of $M$ to get an s-cobordism from $M$ to itself. Trivializing this s-cobordism gives a diffeomorphism $f$ of M to itself such that $(M \times S^1) \# \Sigma$ is the mapping torus of $f$. If you exhibit (as Smale tells us you can) $\Sigma$ as the union of two disks glued by a diffeomorphism $g$ of the boundary, then you get that $f$ is essentially the identity map of $M$ connected sum with $g$.
I'm not sure it exactly answers your question, but Farrell's fibering theorem (Farrell, F. T. The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J. 21 1971/1972 315–346) seems relevant. It implies that a smooth manifold homeomorphic to $M \times S^1$ is diffeomorphic to a fiber bundle over $S^1$, with fiber that is homotopy equivalent to (in fact topologically h-cobordant to) $M$.
• This seems to settle the case of $S^6\times S^1$: since $S^6$ has no exotic structure, construction 1 does not apply. Construction 2 is a special case of 3. By Schulz's paper indicated in Igor's comment, 3. gives as many differential structure as homotopy 7-spheres, i.e. 28 taking orientation into account. By Farrell's theorem these are the only ones. What is still unclear is, say in dimension 8: which exotic 7-spheres $\Sigma$ have the property that $\Sigma\times S^1$ is not diffeomorphic to $S^7\times S^1$ ? – Sylvain Maillot Jan 24 '17 at 16:41
• Also, you are right in assuming I am mostly interested in high dimensions. However just out of curiosity I would like to know whether anything is known for $S^3\times S^1$, apart from the information that comes from Moise and Cerf. For instance, one could conceivably be able to prove that the inertia group is trivial, or is the whole group, without knowing anything about the set of smooth structures on $S^4$... By the way, is it known that $\Theta_4$ is a group? Or should I have said "inertia monoid"? – Sylvain Maillot Jan 24 '17 at 16:49
• For your first comment, if $M' \times S^1$ is diffeomorphic to $M \times S^1$, then $M'$ is h-cobordant to $M$ (pass to the infinite cyclic cover). So for homotopy spheres, of dimension at least 5, $M' \times S^1$ is diffeomorphic to $M \times S^1$, if and only if $M'$ is diffeomorphic to $M$. I don't think anything is known for $S^3 \times S^1$; there's a potential invariant (the Rohlin invariant $\in \mathbb{Z}_2$) but it's not known if it can be nonzero. I don't think it's known if $\Theta_4$ is a group; if the Schoenflies conjecture holds in dimension $4$ there aren't nontrivial inverses – Danny Ruberman Jan 24 '17 at 18:58