I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.

It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;

The function $(M, \partial_{-}M, \partial_{+}M) \mapsto \tau(M): T(\partial_{-}M) \to T(\partial_{+}M)$ extends the modular functor $T$ to a non-degenerate topological quantum field theory.

The proof of the functoriality is unclear for me. I tried to look at Turaev's papers but its harder to understand. Also I don't understand the proof of computation of annomalies (Theorem 4.3 on chapter 4). The method of the proof seems to extend the method of the proof of functoriality to 4-manifolds.

Could you suggest me a textbook or paper etc that explain these theorems or similar material?

Or could you show me more detailed proof of functoriality (and the computation of anomalies) here?

This post imported from StackExchange MathOverflow at 2014-12-14 11:33 (UTC), posted by SE-user Link