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Brown's moduli spaces of
curves and the gravity operad
Clément Dupont and Bruno Vallette |
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Geometry & Topology 21 (2017) 2811–2850
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Abstract |
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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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Keywords
moduli spaces of genus zero curves, operads, mixed Hodge
structures
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Mathematical Subject Classification 2010
Primary: 14H10
Secondary: 14C30, 18D50
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References |
Publication
Received: 12 October 2015
Revised: 26 September 2016
Accepted: 6 November 2016
Published: 15 August 2017
Proposed: Peter Teichner
Seconded: Dan Abramovich, Frances Kirwan
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Authors |
| Clément Dupont | |
| Institut Montpelliérain Alexander
Grothendieck Université de Montpellier Place Eugène Bataillon 34090 Montpellier France |
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| Bruno Vallette | |
| Laboratoire Analyse, Géométrie et
Applications Université Paris 13, Sorbonne Paris Cité CNRS, UMR 7539 93430 Villetaneuse France |
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