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Brown's moduli spaces of curves and the gravity operad

Clément Dupont and Bruno Vallette

Geometry & Topology 21 (2017) 2811–2850

This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown’s moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Brown’s moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergström and Brown which expresses the Betti numbers of Brown’s moduli spaces via the inversion of a generating series. This also generalizes the Salvatore–Tauraso theorem on the nonsymmetric Lie operad.

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moduli spaces of genus zero curves, operads, mixed Hodge structures
Mathematical Subject Classification 2010
Primary: 14H10
Secondary: 14C30, 18D50
Received: 12 October 2015
Revised: 26 September 2016
Accepted: 6 November 2016
Published: 15 August 2017
Proposed: Peter Teichner
Seconded: Dan Abramovich, Frances Kirwan
Clément Dupont
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier
Place Eugène Bataillon
34090 Montpellier
Bruno Vallette
Laboratoire Analyse, Géométrie et Applications
Université Paris 13, Sorbonne Paris Cité
CNRS, UMR 7539
93430 Villetaneuse