This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.



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> Suppose we have two three-dimensional lens spaces $L(n;r)$ and $L(n;s)$ which are homotopy equivalent but not diffeomorphic. Can their products with a circle , $L(n;r) \times S^1$ and $L(n;s) \times S^1$, be diffeomorphic?
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Note that the condition that the lens spaces be homotopy equivalent is necessary because one can lift a diffeomorphism of $L(n;r) \times S^1$ and $L(n;s) \times S^1$ to a homotopy equivalence from $L(n;r) \times \mathbb{R}$ to $L(n;s)\times \mathbb{R}$. Then use the fact that these deformation retract to $L(n;r)$ and $L(n;s)$, respectively.



Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
The answer is no. If $L$, $L^\prime$ are 3-dimensional lens spaces and $S^1\times L$ is diffeomorphic to $S^1\times L^\prime$, then the covering space of $S^1\times L$ corresponding to the torsion subgroup defines an
h-cobordism between $L$ and $L^\prime$ (we have embeddings of L and L′ in the covering space with disjoint images, and the images bound an h-cobordsim). It is an application of Atiyah-Singer fixed point theorem (with contributions by Bott and Milnor), that h-cobordant lens spaces are diffeomorphic. One reference is p.479 in "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications" by Atiyah and Bott.


Various related results and generalizations are discussed in "Toral and exponential stabilization for homotopy spherical spaceforms" by Kwasik and Schultz. Both papers can be easily found online, I think.


