ABSTRACT
Topological Quantum Field Theories (TQFT’S) in dimension 2 + 1 are closely related to representations of mapping class groups of surfaces. This chapter presents some remarks on the spin mapping class groups and their representations arising in the Spin tqft’s constructed by C. The spin mapping class group, denoted by Γσ, is defined as a central extension of the group, denoted by Γ¯σ, of mapping classes preserving a spin structure. After exhibiting a generating set for Γ¯σ the chapter describes the extension Γσ→Γ¯σ in two ways, and observe that it is non-trivial. It then briefly discusses the cobordism categories and the pi-extended spin mapping class groups Γ¯σ. The chapter briefly considers the representations of the Torelli group obtained by restriction, and point out that in the case [k] = 1 these are the classical Birman-Craggs homomorphisms coming from the Rochlin invariant.
