In Segal’s Bourbaki talk on Elliptic cohomology, he mentions offhandedly that:
The set of pairs of commuting elements of a group \(G\) are the set of homotopy classes of maps of a torus into \(BG\).
Semon Rezchikov kindly explained this to me, and I found his explanation so simple and pleasing that I wish to share it.
Recall that \(BG=K(G,1)\) and that \(G = \pi_1(BG)\). A homotopy class of maps \(S^1 \to BG\) is an element of \(G\) (i.e. \(BG\) is the delooping of \(G\)).
[If this statement confuses you, dear reader, Ctrl+F for “Eilenberg-Mac Lane space” in nCategories and Cohomology. If you are uncomfortable with a group as a category, Ctrl+F for “ordinary particle is a point” in From Loop Space Mechanics to Nonabelian Strings.]
Let \((a,b)\) be a pair of commuting elements in \(G\).
In other words, let \((a,b)\) be a pair of paths in \(\pi_1(BG)\) that commute. Note that \(a\) and \(b\) must be loops based at the same point to commute, and that the torus is \(S^1 \times S^1\).
We map the first generator of the torus to \(a\) and the second generator to \(b\).
The first frame of this gif is then \(ab\), and the last frame is \(ba\) (the middle frames are homotopies).
Any paths in \(\pi_1\) that commute (i.e. any pair \((a,b)\) of commuting elements in \(G\)) give a map of the torus into your space \(BG\).