Cohomogeneity one manifolds and selfmaps of nontrivial degree

TL;DR

This paper constructs and analyzes selfmaps of cohomogeneity one manifolds, especially focusing on Lie groups like SU(3), revealing relationships between Weyl group order, Euler characteristic, and selfmap degrees.

Contribution

It introduces a method to construct selfmaps on cohomogeneity one manifolds and extends these to infinite families on SU(3), connecting algebraic and topological properties.

Findings

Constructed selfmaps with computable degrees and Lefschetz numbers.

Established relations between Weyl group order and Euler characteristic.

Realized all possible degrees of selfmaps on SU(3) through compositions.

Abstract

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).