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.](https://mathoverflow.net/questions/48693/smooth-structures-on-the-connected-sum-of-a-manifold-with-an-exotic-sphere) 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](https://mathoverflow.net/questions/158058/two-questions-about-sphere-bundles) The difference is that I am asking about the diffeomorphism type of the total space, not fibered equivalence.



Smooth structures on $M\times S^1$
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) $M$. 


