Weight two compactly supported cohomology of moduli spaces of curves

Abstract

We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves $M_{g,n}$ 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 the weight zero compactly supported cohomology of $M_{g^{\prime },n^{\prime }}$ for $g^{\prime }\le g$ and $n^{\prime }\le 2$, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on $M_g$, including the first such families in odd degrees. In particular, we show that the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k\in \{8,9,11,12,14,15,16,18,19\}$.

arXiv commentv3: 59 pages. Final version, corrected typos. To appear in Duke Math. J

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Cited by in corpus (2)

• Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
• Rational cohomology of ${\mathcal{M}}_{4,1}$