I'd like to discuss a little bit about the problem I asked Diarmuid Crowley that whether all smooth structures on $S^7$ are parallelizable. He first came to using semi-characteristic but then came up with trivial answer which was : Because by chance we know $\pi_7(BO(7))=0$ so all smooth structures are parallelizable. But still I am curious to knpw why smooth structure does not play role here. Can we have trivial bundle in category of topological bundles but not in smooth category?
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See Torsten's comment above for an answer to the question you asked... And here's an answer to a question you didn't ask: Suppose you have homeomorphic manifolds $M^n$ and $\widetilde{M}^n$ that have distinct differentiably structures but are both stably parallelizable (i.e. they embed into a high-dimensional Euclidean space with a trivial normal bundle.) From the same paper that Hatcher cited in his answer to this question, namelt "Vector fields on $\pi$-manifolds" by Bredon and Kosinski, we have the following result: Define a number $\chi^*(M^n)$ to be $\frac{1}{2}\chi(M^n)$ if $n$ is even and $\sum_{i=0}^rH_{i}(M, Z_2)$ (mod 2) for $n = 2r+1$. Then $M^n$ is parallelizable if $n=1,3,7$ or if $1-\chi^*(M) = 1$. (Is this the 'semi-characteristic' you mentioned in your answer?) In particular, since $\chi^*$ is clearly a topological invariant, $M^n$ is parallelizable if and only if $\widetilde{M}^n$ is. This takes care of the stably parallelizable case... I don't know anything about the general case. |
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