2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230554We 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 4.5Algebraic GeometryGeometric Topology14J60, 30F10 (Primary); 14J17 (Secondary)text