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
natanzon@mccme.ru
Anna Pratoussevitch
Dept. of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England
annap@liv.ac.uk
[Abstract-pdf]
\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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