A 3-skeleton for a classifying space for the symmetric group

Abstract

We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg--MacLane classifying space for the symmetric group $\mathfrak{S}_n$. Our complex starts with the presentation for $\mathfrak{S}_n$ with $n-1$ adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of $\mathfrak{S}_n$ in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of $\mathfrak{S}_n$ with untwisted coefficients in $\mathbb{Z}$.