Higher Arf Functions and Moduli Space of Higher Spin Surfacesстатья
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Дата последнего поиска статьи во внешних источниках: 14 сентября 2013 г.
Аннотация: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(d) by a discrete group. Our method is based on a description of an in-spin structure by an m-Arf function, that is a map sigma : pi 1(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 H, (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 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 is an element of 0, 1. (See the results for g = 1 later in the paper.)