Topology of moduli spaces of tropical curves with marked points

Abstract

This article is a sequel to [CGP18]. We study a space ∆ g,n of genus g stable, n-marked tropical curves with total edge length 1. Its rational homology is identified both with top-weight cohomology of the complex moduli space M g,n and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees.We prove a contractibility criterion that applies to various large subspaces of ∆ g,n . From this we derive a description of the homotopy type of ∆ 1,n , the top weight cohomology of M 1,n as an S n -representation, and additional calculations of H i (∆ g,n ; Q) for small (g, n). We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of M g,n in appropriate degrees, by relating both to ∆ g,n . We comment on stability phenomena, or lack thereof.Marked graph complexes. In [CGP18] we gave a cellular chain complex C * (X; Q) associated to any symmetric ∆-complex X, and showed that it computes the reduced rational homology of (the geometric realization of) X. In the case X = ∆ g,n , we are able to deduce that C * (∆ g,n ; Q) is quasi-isomorphic to the marked graph complex G (g,n) , using Theorem 1.1. We shall define G (g,n) precisely in Section 2. Briefly, it is generated by isomorphism classes of connected, loopless, n-marked stable graphs Γ together with the choice of one of the two possible orientations of R E (Γ) . The main difference between G (g,n) and the hairy graph complexes in the existing literature [CKV13, CKV15, KW Ž16, TW17] is that we consider graphs with n labeled markings, rather than n unlabeled hairs or half-edges. 1 We use the terminology "marked" rather than "hairy" for this reason. Kontsevich's graph complex G (g) [Kon93, Kon94] occurs as the special case n = 0.Theorem 1.4. For 2g − 2 + n > 0, there is a natural split injection of chain complexes1 We expect that there may be natural maps between the S n -coinvariants of our marked graph complex and some existing hairy graph complexes, but have been unable to find such precise relations.

Filtrations of moduli spaces of tropical weighted stable curves

“…In Example 5.10, we use this Theorem to compute the Top Weight Cohomology of M 1,3 , confirming results obtained in [CGP19].…”

“…In Section 4, we will also define also graph complexes G (g,A) . These objects generalize the graph complexes G (g,n) defined in section 2.4 of [CGP19]. We deduce a Filtration Theorem analogous to the Theorem A for these complexes (see Section 4 for further details).…”

“…We deduce a Filtration Theorem analogous to the Theorem A for these complexes (see Section 4 for further details). In order to study further these graph complexes and their homology, in Section 5 we extend to ∆ g,A the theory developed in [CGP21] and [CGP19] for ∆ g,n , treating it as a symmetric ∆-complex for every A, and then we generalize Theorem 1.4 of [CGP19] showing that there is a natural surjection of chain complexes C * (∆ g,A ) → G (g,A) decreasing degrees by 2g − 1, inducing isomorphisms on homology…”

“…In [Yun21], there are computations for the S n -equivariant rational homology of the tropical moduli spaces ∆ 2,n for n ≤ 8. Sections 5 and 6 develop on tools and language introduced in [CGP21] and [CGP19], which we adapt to study ∆ g,A .…”

“…This result is generalized for g ≥ 1 and for all A in [KLSY20] by Kannan, Li, Yun and the author. • In [CGP21] and [CGP19] Chan, Galatius and Payne give results on the top weight cohomology of ∆ g,n . In [KLSY20] similar results are given for ∆ g,A .…”