Generalised spin structures on 2-dimensional orbifolds

TL;DR

This paper studies generalized spin structures on 2D orbifolds, focusing on hyperbolic cases, and explores their existence, symmetry actions, and implications for moduli spaces of contact structures.

Contribution

It characterizes the existence conditions for r-spin structures on hyperbolic orbifolds and analyzes the diffeomorphism group action on these structures.

Findings

Conditions for existence of r-spin structures established

Number and size of diffeomorphism orbits determined

Application to moduli space of taut contact circles provided

Abstract

Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold \Sigma are r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle ST\Sigma. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomorphism group of \Sigma on the set of r-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the 3-dimensional geometry \tilde{SL}_2.