June 21: Universal and unitary TQFTs

Kyle Ormsby

TQFTs assign numerical invariants to smooth manifolds. Following Freedman et al and Kreck-Teichner, we will discuss the power of these invariants. In dimensions d at least 4, the universal pairing on d-manifolds with boundary is non-positive, from which it follows that unitary d-dimensional TQFTs cannot detect smooth structures. We also discuss a technique for proving positivity of the universal pairing that lifts this obstruction to the detection of smooth structures in dimensions 0, 1, 2, and 3. (The case d = 3 is due to Calegari-Freedman-Walker.)