Parity of the spin structure defined by a quadratic differential

Lanneau, Erwan

doi:10.2140/gt.2004.8.511

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Parity of the spin structure defined by a quadratic differential

Erwan Lanneau

Geometry & Topology 8 (2004) 511–538

DOI: 10.2140/gt.2004.8.511

arXiv: math.GT/0210116

Abstract

According to the work of Kontsevich–Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities, is the parity of the spin structure.

We show that for the moduli space of quadratic differentials, the spin structure is constant on every stratum where it is defined. In particular this disproves the conjecture that it classifies the non-hyperelliptic connected components of the strata of quadratic differentials with prescribed singularities. An explicit formula for the parity of the spin structure is given.

Keywords

quadratic differentials, Teichmüller geodesic flow, moduli space, measured foliations, spin structure

Mathematical Subject Classification 2000

Primary: 32G15

Secondary: 30F30, 30F60, 58F18

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Publication

Received: 29 July 2003

Revised: 12 March 2004

Accepted: 16 December 2004

Published: 13 March 2004

Proposed: Benson Farb

Seconded: Jean-Pierre Otal, Shigeyuki Morita

Authors

Erwan Lanneau
Institut de mathématiques de Luminy Case 907, 163 Avenue de Luminy F-13288 Marseille Cedex 9 France

Volume 8, issue 2 (2004)