The only topological information in 3-manifolds besides homology and homotopy is Reidemeister torsion (see this question). TQFT sees Ray-Singer torsion, which is the same thing. Indeed, it was this discovery by Schwartz (and independently, unpublished, by Singer) which sparked the subject and which motivated Witten's work on a TQFT for the Jones polynomial.
Daniel Moskovich
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