An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by cutting and pasting.

**I am curious if TQFTs give a complete set of invariants of manifolds (either in topological category or smooth category).** My impression is that the locality of TQFTs, which enables the computation via cutting and pasting, seems like quite a strong property, so it might be possible that there are two manifolds which cannot be distinguished by any TQFTs.