Infiniteness of $A_\infty$-types of gauge groups

TL;DR

This paper proves that the collection of gauge groups of principal G-bundles over spheres exhibits infinitely many distinct $A_ infty$-types, contrasting with the finite types observed for finite complexes.

Contribution

It demonstrates the infiniteness of $A_ infty$-types of gauge groups over spheres, extending previous finiteness results for finite complexes.

Findings

Number of $A_ infty$-types is infinite over spheres.

Finiteness of $A_n$-types for finite complexes when $n< infty$.

Gauge groups over spheres have infinitely many distinct homotopy types.

Abstract

Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is proved by Crabb--Sutherland and the second author that the number of $A_n$-types of the gauge groups of $P$ is finite if $n<\infty$ and $K$ is a finite complex. We show that the number of $A_\infty$-types of the gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely many $P$.