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.