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 $\mathbb{Z}$.

Figures (4)

• Figure 1: The three kinds of $2$-cells in $X$. Here we label edges with "$i$" instead of "$e_i$", etc. The condition that $i<j$ is part of the definition for $d_{ij}$.
• Figure 2: The $3$-cells of $X$, except for $c^{37}_i$, which is in another figure. Again we label edges with "$i$" instead of "$e_i$". The condition that $i<j<k$ is part of the definition for $c^{33}_{ijk}$. The cells $c^{32}_{ij}$ and $c^{34}_{ij}$ are illustrated with the assumption that $i<j$, but there are also versions of these cells where the reverse inequality holds.
• Figure 3: The $3$-cell $c^{37}_i$, with faces labeled by the key here. Again we label edges with "$i$" instead of "$e_i$".
• Figure 4: An example of a filling of $\psi_2(\partial^Q_3 \sigma)$, as in Proposition \ref{['pr:boundcellbbcase1']}, for $\sigma=[s_j|\rho(i,j)|\rho(k,j)])$. Here $k=1$, $i=4$, and $j=7$. Each edge $e_i$ is labeled with "$i$". The left figure is the visualization of $\psi_2(\partial^Q_3 \sigma)$ as a cuboid tiled by squares and hexagons. The right figure is the same cuboid, exploded into $3$-cells of types $c^{33}_{ijk}$, $c^{34}_{ij}$, and $c^{37}_i$.

Theorems & Definitions (143)

• Theorem 1.1
• Proposition 1.2
• proof
• Theorem 1.3
• proof
• Definition 1.4: The complex $P_*$
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Remark 2.3