Unstable cohomology of $\mathsf{GL}_{2n}(\mathbb{Z})$ and the odd commutative graph complex

Abstract

We study a closed differential form on the symmetric space of positive definite matrices, which is defined using the Pfaffian and is $\mathsf{GL}_{2n}(\mathbb{Z})$ invariant up to a sign. It gives rise to an infinite family of unstable classes in the compactly-supported cohomology of the locally symmetric space for $\mathsf{GL}_{2n}(\mathbb{Z})$ with coefficients in the orientation bundle. Furthermore, by applying the Pfaffian forms to the dual Laplacian of graphs, and integrating them over the space of edge lengths, we construct an infinite family of cocycles for the odd commutative graph complex. By explicit computation, we show that the first such cocycle gives a non-trivial class in $H^{-6}(\mathsf{GC}_3)$.