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Journal of Lie Theory 19 (2009), No. 1, 107--148
Copyright Heldermann Verlag 2009

Higher Arf Functions and Moduli Space of Higher Spin Surfaces

Sergey Natanzon
Moscow State University, Korp. A - Leninske Gory, 11899 Moscow, Russia
and: Inst. of Theoretical and Experimental Physics, Independent University of Moscow, Bolshoi Vlasevsky Pereulok 11, 119002 Moscow, Russia

Anna Pratoussevitch
Dept. of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England


\def\Z{{\Bbb Z}} We describe all connected components of the space of pairs $(P,s)$, where $P$ is a hyperbolic Riemann surface with finitely generated fundamental group and $s$ is an $m$-spin structure on $P$. We prove that any connected component is homeomorphic to a quotient of ${\mathbb R}^d$ by a discrete group.\endgraf Our method is based on a description of an $m$-spin structure by an $m$-Arf function, that is a map $\sigma\colon\pi_1(P,p)\to {\Z}/m{\Z}$ with certain geometric properties. We prove that the set of all $m$-Arf functions has a structure of an affine space associated with $H_1(P,{\Z}/m{\Z})$. We describe the orbits of $m$-Arf functions under the action of the group of homotopy classes of surface autohomeomorphisms. Natural topological invariants of an orbit are the unordered set of values of the $m$-Arf functions on the punctures and the unordered set of values on the $m$-Arf-function on the holes. We prove that for $g>1$ the space of $m$-Arf functions with prescribed genus and prescribed (unordered) sets of values on punctures and holes is either connected or has two connected components distinguished by the Arf invariant $\delta\in\{0,1\}$. Results for $g=1$ are also given.

Keywords: Higher spin surfaces, Arf functions, lifts of Fuchsian groups.

MSC: 14J60, 30F10; 14J17, 32S25

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