Determination of the multiplicative nilpotency of self-homotopy sets

TL;DR

This paper investigates the nilpotency of semigroups formed by self-homotopy maps that induce trivial homomorphisms on homotopy groups, focusing on finite complexes, Lie groups, and Hopf spaces, revealing their nilpotency properties.

Contribution

It determines the nilpotency of these semigroups for specific classes of spaces, including compact Lie groups and finite Hopf spaces, and explores how nilpotency varies with group rank.

Findings

Semigroup of self-homotopy maps is nilpotent for finite complexes.

Nilpotency varies with the rank of Lie groups.

Constructs Lie groups with arbitrarily large nilpotency.

Abstract

The semigroup of the homotopy classes of the self-homotopy maps of a finite complex which induce the trivial homomorphism on homotopy groups is nilpotent. We determine the nilpotency of these semigroups of compact Lie groups and finite Hopf spaces of rank 2. We also study the nilpotency of semigroups for Lie groups of higher rank. Especially, we give Lie groups with the nilpotency of the semigroups arbitrarily large.