Weight 2 compactly supported cohomology of moduli spaces of curves

We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves Mg,n and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For n=0, we express this cohomology in terms of W0Hc∙(Mg′,n′) for g′≤g and n′≤2, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on Mg, including the first such families in odd degrees. In particular, we show that dimH4g−k(Mg) grows at least exponentially with g for k∈{8,9,11,12,14,15,16,18,19}.