# 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 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 $$BG$$.