Journal of Lie Theory

no. 1
Higher Arf Functions and Moduli Space of Higher Spin Surfaces
Sergey Natanzon1, 2; Anna Pratoussevitch3
1Moscow State University, Korp. A - Leninske Gory, 11899 Moscow, Russia
2Inst. of Theoretical and Experimental Physics, Independent University of Moscow, Bolshoi Vlasevsky Pereulok 11, 119002 Moscow, Russia
3Dept. of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England
Journal of Lie Theory, Volume 19 (2009) no. 1, pp. 107-148
\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.
DOI: 10.5802/jolt.544
Classification: 14J60, 30F10, 14J17, 32S25
Keywords: Higher spin surfaces, Arf functions, lifts of Fuchsian groups

Sergey Natanzon  1 , 2 ; Anna Pratoussevitch  3

1 Moscow State University, Korp. A - Leninske Gory, 11899 Moscow, Russia
2 Inst. of Theoretical and Experimental Physics, Independent University of Moscow, Bolshoi Vlasevsky Pereulok 11, 119002 Moscow, Russia
3 Dept. of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England 
Sergey Natanzon; Anna Pratoussevitch. Higher Arf Functions and Moduli Space of Higher Spin Surfaces. Journal of Lie Theory, Volume 19 (2009) no. 1, pp. 107-148. doi: 10.5802/jolt.544
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