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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cyclic operad formality for compactified moduli spaces of genus zero surfaces


Authors: Jeffrey Giansiracusa and Paolo Salvatore
Journal: Trans. Amer. Math. Soc. 364 (2012), 5881-5911
MSC (2010): Primary 18D50; Secondary 55P48, 14H15, 81Q30, 81T45
Published electronically: May 21, 2012
MathSciNet review: 2946936
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Abstract: The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.


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Additional Information

Jeffrey Giansiracusa
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom
Email: j.h.giansiracusa@gmail.com

Paolo Salvatore
Affiliation: Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
Email: salvator@mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05553-X
PII: S 0002-9947(2012)05553-X
Keywords: Cyclic operad, framed little discs, moduli of curves, operad formality, graph complex
Received by editor(s): September 10, 2010
Received by editor(s) in revised form: January 25, 2011
Published electronically: May 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.