The only topological information in 3-manifolds besides homology and homotopy is Reidemeister torsion (see <a href="http://mathoverflow.net/questions/47380/fundamental-group-and-complete-invariant-of-irreducible-3-manifolds">this question</a>). TQFT sees <a href="http://en.wikipedia.org/wiki/Analytic_torsion">Ray-Singer torsion</a>, which is the same thing. Indeed, it was this discovery <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=676337&loc=fromrevtext">by Schwartz</a> which sparked the subject and which motivated Witten's work on a <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=990772&loc=fromrevtext">TQFT for the Jones polynomial</a>.