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

Abstract

We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.

32 pages, 6 figures; v3: exposition improved, typos corrected; v4: Lemma 3.9 corrected; v5: small changes in Def. 4.2 and proof of Lemma 4.5