Multivalued sections and self-maps of sphere bundles

TL;DR

This paper proves that for certain finite group actions on sphere bundles, there always exists a $G$-equivariant map from any principal $G$-bundle over a compact ENR to the sphere in a related orthogonal module, extending previous work on self-maps.

Contribution

It establishes the existence of $G$-maps from principal $G$-bundles to sphere bundles under specific fixed point conditions, generalizing prior results on self-maps of sphere bundles.

Findings

Existence of $G$-maps from principal $G$-bundles to sphere bundles.

Conditions on the $G$-module $V$ related to fixed points and dimensions.

Extension of previous work on self-maps of sphere bundles.

Abstract

Let $G$ be a finite group and $V$ a finite dimensional (non-zero) orthogonal $G$-module such that, for each prime $p$ dividing the order of $G$, the subspace of $V$ fixed by a Sylow $p$-subgroup of $G$ is non-zero and, if the dimension of $V$ is odd, has dimension greater than $1$. Using ideas of Avvakumov, Karasev, Kudrya and Skopenkov and work of Noakes on self-maps of sphere bundles, we show that, for any principal $G$-bundle $P\to X$ over a compact ENR $X$, there exists a $G$-map from $P$ to the unit sphere $S(V)$ in $V$.