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 W 0 H • c (M g ′ ,n ′ ) for g ′ ≤ g and n ′ ≤ 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 dim H 4g−k (M g ) grows at least exponentially with g, for k ∈ {8,

The weight two compactly supported Euler characteristic of moduli spaces of curves

“…Let g and n be non-negative integers. The graph complex G pg,nq studied in [5,19] is generated by connected graphs of genus g with no loop edges and no vertices of valence 2. The vertices of valence 1 are called external, and the other vertices are internal.…”

“…The graph complexes X g,n and weight two compactly supported cohomology of M g,n . The main result of [19] is the identification of gr 2 H ‚ c pM g,n q with the cohomology of a graph complex X g,n that is close to graph complexes arising in knot theory and the embedding calculus. We shall use this identification to show our main Theorem 1.1, by computing the Euler characteristic of X g,n .…”

“…We refer to [19] for details. Theorem 3.5 (Theorem 1.1 of [19]). There is an S n -equivariant isomorphism HpX g,n q -gr 2 H c pM g,n q for each pg, nq ‰ p1, 1q with 2g `n ě 3.…”

“…Algebraic geometry techniques have recently yielded new nonvanishing results and lower bounds on dimensions of cohomology groups in the unstable range [4,5,19]. The algebraic structure of M g,n as a Deligne-Mumford stack endows its rational cohomology with a mixed Hodge structure.…”

“…This is especially striking since the stable cohomology grows subexponentially and, at the time of their writing and for nearly twenty years after, there were no known examples where H k pM g q is nonzero for odd k. The first such example was H 5 pM 4 q [22]. All other known examples are proved using the associated graded of the weight filtration and graph complex techniques; see [4,19]. Even if one is primarily interested in the cohomology of M g , for inductive arguments using the boundary structure of the moduli space of stable curves as in [1,9], it is essential to understand the cohomology of moduli spaces of curves with marked points M g,n , and to understand these not only as vector spaces but as representations of the symmetric group S n , with the action induced by permuting marked points.…”