Witten's Invariant of 3-Dimensional Manifolds: Loop Expansion and Surgery Calculus

We review two different methods of calculating Witten's invariant: a stationary phase approximation and a surgery calculus. We give a detailed description of the 1-loop approximation formula for Witten's invariant and of the technics involved in deriving its exact value through a surgery construction of a manifold. Finally we compare the formulas produced by both methods for a 3-dimensional sphere S^3 and a lens space L(p,1).