Equivariant vector fields on spheres

Abstract:

We address the following question: If $G$ is a compact Lie group and $S(M)$ is the unit sphere of an $R[G]$-module $M$, then how many orthonormal $G$-invariant vector fields can be found on $S(M)$? We call this number the $G$-field number of $M$. Under reasonable hypotheses on $M$, we reduce this question to determining when the difference of two $G$-vector bundles vanishes in a certain subquotient of the $K{O_G}$-theory of a real projective space. In this general setting, we solve the problem for $2$-groups, for odd-order groups, and for abelian groups. If $M$ also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on $M$, the $G$-field number depends only on the dimensions of the fixed point sets of $M$.