Higher Arf Functions and Topology of the Moduli Space of Higher Spin Riemann Surfaces

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 R by a discrete group. Our method is based on a description of an m-spin structure by an m-Arf function, that is a map σ : π1(P, p) → Z/mZ with certain geometric properties. We prove that the set of all m-Arf functions has a structure of an affine space associated with H1(P,Z/mZ). 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 mArf-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 δ ∈ {0, 1}. (See the results for g = 1 later in the paper.)