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One knows that the tangent bundle of an exotic sphere is bundle isomorphic to the tangent bundle of the standard sphere.

Are there closed manifolds that admit several different differentiable structures in which two of the tangent bundles are not bundle isomorphic?

lavinia
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1 Answers1

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In Corollary 1 of this paper, Milnor constructs two smooth manifolds $M_1, M_2$ that are homeomorphic but have $TM_1$ trivial and $TM_2$ not even stably trivial. The proof involves constructing a bundle that's stably trivial ($s$-trivial in the notation of the paper) in the category of microbudles but not stably trivial in the nicer category of vector bundles.

anomaly
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  • One thing to note: the manifolds in the paper don't seem to be compact. I don't know whether that's a crucial part of the construction; unfortunately, I don't know much about microbundles beyond their definition and a few results like this one. – anomaly Nov 21 '17 at 20:06
  • So maybe for closed manifolds the tangent bundle is unique. – lavinia Nov 22 '17 at 01:48
  • I would suspect it's not even true for the compact case, but I don't have a counterexample. In the opposite direction, one possible place to start is the result that for a closed smooth manifold, there are only finitely many distinct smooth strutures. – anomaly Nov 22 '17 at 04:00
  • @anomaly Are $TM_1$ and $TM_2$ homeomorphic? – Anubhav Mukherjee Dec 06 '17 at 15:17
  • @AnubhavMukherjee: The total spaces? Probably not, since $TM_1$ is trivial. – anomaly Dec 06 '17 at 15:22
  • @anomaly Yes I am talking about total space. I agree that $TM_1$ is trivial, but that doesnot effect their homeomrphism type, is it? – Anubhav Mukherjee Dec 06 '17 at 15:24
  • @AnubhavMukherjee: It doesn't, but it makes a bit suspicious that they might be homeomorphic. I wouldn't bet any money on it, though, and I don't have a proof in either direction. – anomaly Dec 06 '17 at 15:29
  • @anomaly It would be ineresting, if they are homeomorphic. Then we can understand the obstruction of converting a homeomorphism into bundle isomorphism. Don't you think? – Anubhav Mukherjee Dec 06 '17 at 15:32
  • @anomaly Actually they are homeomorphic. I read the paper. It's a nice one. Thanks for sharing. – Anubhav Mukherjee Dec 07 '17 at 00:23
  • What was the Milnor paper? The link is dead. – Mr. Brown Jul 19 '23 at 19:21