0
$\begingroup$

The question is rather open-ended but I hope it is concrete enough.

If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what are the obstructions to $M/G$ being parallelizable?

This question was motivated by looking at the Hopf fibrations $S^1\to S^3\to S^2$ and $S^3\to S^7\to S^4$, where the total space is parallelizable but the base are not. It would also be nice to see what are these obstructions in the cases of the fibrations $O(n)\to O(n+1)\to S^n$ and how do they change depending on $n$.

Any insights are welcome. Thank you very much.

$\endgroup$
3
  • $\begingroup$ Wouldn't these just be the obstructions to framing an arbitrary vector bundle, i.e. the unreduced Whitney classes? Or are you interested in relating the obstructions for the total space with the obstruction for the quotient space? $\endgroup$ Nov 21, 2022 at 18:21
  • $\begingroup$ @RyanBudney Precisely, is there a way to relate those? For example, understanding where they appear in the Serre spectral sequence for M\to M/G\to BG. $\endgroup$ Nov 21, 2022 at 19:09
  • $\begingroup$ Sure, there is a relation. I have not worked out the details, but if you take the obstruction-theoretic point of view there is a direct relation via the fiberwise cell structure on the total space -- the one that fits with the cell structure of the group and the cell structure of the base space. At that level, the obstructions will all interact nicely with each other, and the local structure of the group action. So you'll see obstructions for the vector bundle neighbouhoods of orbits, etc. $\endgroup$ Nov 21, 2022 at 19:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.