On the Groupoid of Transformations of Rigid Structures on Surfaces

We prove that the 2-groupoid of transformations of rigid structures on surfaces has a finite presentation, establishing a result first conjectured by Moore and Seiberg. We also show that a finite dimensional, unitary, cyclic topological quantum field theory gives rise to a representation of this 2-groupoid. AMS Classification: 57 N 10, 57 M 25, 16 A 46. Keywords: Mapping class groups, rigid structures, duality groupoid, DAP decompositions, TQFT. Contents 1 Introduction 2 2 2-groupoids 3 2.1 Algebraic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The geometric point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 The Moore-Seiberg equations 7 3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Topological interpretation of the generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . 8..