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Torsion in classifying spaces of gauge groups
Part of
- Fiber spaces and bundles
Masaki Kameko
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View

Published online by Cambridge University Press:

01 April 2024, pp. 1-16
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We determine when the integral homology of the classifying space of a $PU(n)$-gauge group over the sphere $S^2$ has torsion.

3 - Lifting Geometry to Mapping Spaces I: Lie Groups
Alexander Schmeding, Nord Universitet, Norway
Book:

An Introduction to Infinite-Dimensional Differential Geometry

Published online:

08 December 2022

Print publication:

22 December 2022, pp 48-79
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Summary

In this chapter, one aim is to study spaces of mappings taking their values in a Lie group. It will turn out that these spaces carry again a natural Lie group structure. However, before we prove this, the definition and basic properties of (infinite dimensional) Lie groups and their associated Lie algebras are recalled. Infinite-dimensional Lie theory (beyond Banach spaces) is by comparison relatively young and in its modern form goes back to Milnor’s seminal works. One key feature of infinite-dimensional Lie theory is that the conncection between Lie algebra and Lie group is looser then in finite dimensions. For advanced tools in Lie theory one has to require the Lie group to be regular (in the sense of Milnor). These concepts are introduced and considered for several main classes of examples, such as the diffeomorphism groups, loop groups and gauge groups.

The mod-p homology of the classifying spaces of certain gauge groups
Part of
- Fiber spaces and bundles
- Quantum field theory; related classical field theories
Daisuke Kishimoto, Stephen Theriault
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 6 / December 2023

Published online by Cambridge University Press:

19 September 2022, pp. 1805-1817

Print publication:

December 2023
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Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.