Duke Mathematical Journal

Moduli spaces and formal operads

V. Navarro, P. Pascual, A. Roig, and F. Guillén Santos

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Let M̲g,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations that relate the different moduli spaces identifying marked points, the family (M̲g,l)g,l is a modular operad of projective smooth Deligne-Mumford stacks M̲. In this paper, we prove that the modular operad of singular chains S*(M̲;Q) is formal, so it is weakly equivalent to the modular operad of its homology H*(M̲;Q). As a consequence, the up-to-homotopy algebras of these two operads are the same. To obtain this result, we prove a formality theorem for operads analogous to the Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.

Article information

Duke Math. J., Volume 129, Number 2 (2005), 291-335.

First available in Project Euclid: 27 September 2005

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Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic) 18D50: Operads [See also 55P48]


Santos, F. Guillén; Navarro, V.; Pascual, P.; Roig, A. Moduli spaces and formal operads. Duke Math. J. 129 (2005), no. 2, 291--335. doi:10.1215/S0012-7094-05-12924-6. https://projecteuclid.org/euclid.dmj/1127831440

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