Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way.


I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are.


Which is the smallest dimension in which one can find such examples?


What if I ask the same question for $k$ pairwise non-diffeomorphic manifolds?


Can we have $k=\infty$?



Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle
Here are examples of non-diffeomorphic **closed** manifolds with diffeomorphic
tangent bundles:


1. 3-dimensional lens spaces have trivial tangent bundles, which are diffeomorphic if and only if the lens spaces are homotopy equivalent, e.g. $L(7,1)$, $L(7,2)$ are not homeomorphic, but their tangent bundles are diffeomorphic. This follows from proofs in [Milnor, John, Two complexes which are homeomorphic but combinatorially distinct.
Ann. of Math. (2) 74 1961 575--590].
2. Tangent bundle to any exotic $n$-sphere is diffeomorphic to $TS^n$ as proved in [De Sapio, Rodolfo, Disc and sphere bundles over homotopy spheres, Math. Z. 107 1968 232--236].
3. In dimensions $n\ge 5$ one can attack this question via surgery theory. For example, let $f:N\to M$ be a homotopy equivalence of closed $n$-manifolds that has trivial normal invariant (which is a bit more than requiring that $f$ preserves stable tangent bundle). Multiply $f$ by the identity map of $(D^n, S^{n-1})$, where $D^n$ is the closed $n$-disk. Then Wall's $\pi-\pi$ theorem implies that $M\times D^n$ and $N\times D^n$ are diffeomorphic, so if tangent bundles of $M, N$ are trivial, this gives manifolds with diffeomorphic tangent bundles.
4. To illustrate the method in 3, here is a particular example of **infinitely many** non-homeomorphic closed manifolds with diffeomorphic tangent bundles. Fix any closed $(4r-1)$-manifold $M$ where $r\ge 2$ such that $TM$ is stably trivial and $\pi\_1(M)$ has (nontrivial) elements of finite order. Then results of Chang-Weinberger imply that there are infinitely many closed $n$-manifolds $M\_i$ that are simply-homotopy equivalent to $M$ and such that the tangent bundles $TM\_i$ are all diffeomorphic (I am not quite sure how to get them be diffeomorphic to $M\times\mathbb R^n$ even though this should be possible). I know how to deduce this from [On invariants of Hirzebruch and Cheeger-Gromov, Geom. Topol. 7 (2003), 311--319].
5. I have been thinking extensively of related issues, so you might want to look at my papers at arxiv, e.g. [this one.](http://arxiv.org/abs/0912.4869)


