The mod-$p$ homology of the classifying spaces of certain gauge groups

TL;DR

This paper computes the mod-$p$ homology of classifying spaces of certain gauge groups over $S^4$, revealing new algebraic topological properties under specific conditions on the Lie group and prime $p$.

Contribution

It provides explicit calculations of the mod-$p$ homology for classifying spaces of gauge groups associated with simply-connected simple compact Lie groups, under particular restrictions.

Findings

Explicit mod-$p$ homology calculations for classifying spaces $B\,\mathcal{G}_k$.

Conditions $n_{\ell}<p-1$ and $p \nmid k$ are crucial for the results.

Enhances understanding of the algebraic topology of gauge groups and their classifying spaces.

Abstract

Let $G$ be a simply-connected, simple compact Lie group of type $\{n_{1},\ldots,n_{\ell}\}$, where $n_{1}\le\cdots \le n_{\ell}$. Let $\mathcal{G}_k$ be the gauge group of the principal $G$-bundle (namedright{P}{}{S^{4}}) whose isomorphism class is determined by the the second Chern class having value $k\in\mathbb{Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal{G}_k$ provided that $n_{\ell}<p-1$ and $p$ does not divide $k$.