Real forms of higher spin structures on Riemann orbifolds

• Main Author: Riley, Heather
• Published: University of Liverpool 2015
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.677521

In this thesis we study the space of m-spin structures on hyperbolic Klein orbifolds. A hyperbolic Klein orbifold is a hyperbolic 2-dimensional orbifold with a maximal atlas whose transition maps are either holomorphic or anti holomorphic. Hyperbolic Klein orbifolds can be described as pairs (P,\tau), where P is a quotient of the hyperbolic plane by a Fuchsian group \Gamma and \tau an anti-holomorphic involution on P. An m-spin structure on a hyperbolic Klein orbifold P is a complex line bundle L such that the m-th tensor power of L is isomorphic to the cotangent bundle of P and L is invariant under the involution \tau. We only consider a certain class of hyperbolic Klein orbifolds which we call nice Klein orbifolds, namely those where no fixed points of the involution \tau are fixed by any elements of the Fuchsian group \Gamma. We describe topological invariants of m-spin structures on nice Klein orbifolds and determine the conditions under which such m-spin structures exist. We describe all connected components of the space of m-spin structures on nice Klein orbifolds and prove that any connected component is homeomorphic to a quotient of \mathbb{R}^d by a discrete group.