Geometry & Topology, Vol. 8 (2004) Paper no. 12, pages 511--538.

Parity of the spin structure defined by a quadratic differential

Erwan Lanneau

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, Teichmueller geodesic flow, moduli space, measured foliations, spin structure

AMS subject classification. Primary: 32G15, 30F30, 30F60. Secondary: 58F12, 57R15.

DOI: 10.2140/gt.2004.8.511

E-print: arXiv:math.GT/0210116

Submitted to GT on 29 July 2003. (Revised 12 March 2004.) Paper accepted 16 December 2004. Paper published 13 March 2004.

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

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