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Forum Mathematicum

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Maslov index, lagrangians, mapping class groups and TQFT

1Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

2Institut de Mathématiques de Jussieu (UMR 7586 CNRS), Case 247, 4 pl. Jussieu, 75252 Paris Cedex 5, France

Citation Information: Forum Mathematicum. Volume 25, Issue 5, Pages 1067–1106, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/form.2011.143, August 2011

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Given a mapping class f of an oriented surface and a lagrangian λ in the first homology of , we define an integer . We use to describe a universal central extension of the mapping class group of as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.

Keywords: Signature cocycle; extended manifold; mapping class group; Maslov index; TQFT

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