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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.

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?

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.

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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.

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    $\begingroup$ Related: one can use Seiberg–Witten/Heegaard Floer homology to show that two lens spaces are homology cobordant if and only if they are diffeomorphic. This is explained with some references (as well as with a new proof) here: arxiv.org/abs/1505.06970 . $\endgroup$
    – Marco Golla
    Commented May 5, 2021 at 18:38
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    $\begingroup$ @MarcoGolla: thank you, I was unaware of this paper (even though I knew that d-invariants distinguish homology cobordsim classes of lens spaces). Perhaps, I should mention that Atiyah-Singer-Bott-Milnor argument works for lens spaces of any dimension (not just dimension 3). $\endgroup$
    – Igor Belegradek
    Commented May 5, 2021 at 18:49
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    $\begingroup$ I really like the idea of lifts giving an H-cobordism. Is this a standard argument, or something you thought of for this post? $\endgroup$
    – Jason DeVito - on hiatus
    Commented May 5, 2021 at 20:52
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    $\begingroup$ @JasonDeVito: this is a standard argument (or rather, if two closed manifolds cross $\mathbb R$ are differeomorphic, this gives an h-cobordsim); the lift part is rather trivial. A good reference is Milnor's paper on Hauptvermutung, "Two complexes which are homeomorphic but combinatorially distinct", which you may enjoy reading. $\endgroup$
    – Igor Belegradek
    Commented May 5, 2021 at 21:12
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    $\begingroup$ Yes, the lifting part is trivial. Sorry for not being clearer in my question, but you answered exactly what I meant to ask. And it's always a pleasure to read Milnor. Thanks! $\endgroup$
    – Jason DeVito - on hiatus
    Commented May 5, 2021 at 21:42

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