Toots for Diffgeometer1@mathstodon.xyz account

Written by Diffgeometer1 on 2024-10-04 at 01:57

Let (\pi: SO(n+1)\rightarrow S^n) be the map given by[\pi(P)=Px]where (x\in S^n) is some fixed element of (S^n). Then (\pi: SO(n+1)\rightarrow S^n) is a principal (SO(n))-bundle. Assume (n>2). However, my intuition is that this cannot be a trivial principal bundle since it would imply[SO(n+1)\simeq S^n\times SO(n)]which I don't think is right. I don't know what the de Rham cohomology of (SO(n)) is. If I can show that the left and right sides have different de Rham cohomology groups that would prove that (\pi: SO(n+1)\rightarrow S^n) cannot be a trivial principal bundle. Does anyone have a simple way of showing that (SO(n+1)) cannot be homeomorphic to (SO(n)\times S^n)? Also I want to avoid the use of characteristic classes or (K)-theory. I want to limit things to homology, cohomology, and fundamental groups only.

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Written by Diffgeometer1 on 2024-09-25 at 18:06

The index of a vector field is defined in terms of isolated zeros of the vector field.

If a vector field is nonvanishing, is the index defined to be zero trivially since it has no zeros and there is nothing to calculate?

A related question: does a manifold always admit a vector field with isolated zeros?

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