Finiteness of $A_n$-equivalence types of gauge groups

TL;DR

This paper proves that the number of gauge groups of principal G-bundles over a finite CW complex is finite up to $A_n$-equivalence for finite n, with specific bounds provided for SU(2) over S^4.

Contribution

It establishes the finiteness of $A_n$-equivalence types of gauge groups over finite CW complexes, providing explicit bounds for certain cases.

Findings

Number of gauge groups is finite up to $A_n$-equivalence for finite n.

Provides a lower bound for $A_n$-equivalence types of gauge groups of SU(2)-bundles over S^4.

Demonstrates finiteness results in the classification of gauge groups.

Abstract

Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show that the number of gauge groups of principal $G$-bundles over $B$ is finite up to $A_n$-equivalence for $n<\infty$. As an example, we give a lower bound of the number of $A_n$-equivalence types of gauge groups of principal $\SU(2)$-bundles over $S^4$.