Higher Arf Functions and Topology of the Moduli Space of Higher Spin   Riemann Surfaces
  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. 
Sergey Natanzon and Anna Pratoussevitch