Maps of a Torus

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 CohomologyIf 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 $b$.

Source
Source

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 $BG$.

Notes on Covering Spaces as Extensions

This post assumes knowledge of fiber bundles, the group action functor, groupoids, and basic vector calculus. I am in the process of learning the topics discussed below, and I deeply appreciate constructive feedback. 

How does a big space cover a little one?

Given a covering space $E \to B$ we can uniquely lift any path in the base space (once you choose a starting point) to a path in $E$. Conversely, we can create a covering space of $B$ by letting the fiber over $b$ be $F(b)$.

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