Journal of Lie Theory

\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.