In this talk, I will explain how in any semisimple (n+1)-dimensional anomalous TQFT, Hopf linking leads to a perfect pairing between (certain equivalence classes of) operators/defects of dimension k and of dimension (n-k). In fact, I will outline how non-degeneracy of these `S-matrix pairings' precisely characterizes the anomalous TQFTs amongst all relative (n+2)/(n+1)-dimensional TQFTs.
Along the way, I will explain how the set of equivalence classes of codimension-k operators may be thought of as the k-th `homotopy set' pi_k T of a TQFT T — for k >= 1, this set forms the basis of a certain kind of fusion ring, generalizing the group structure familiar from homotopy theory.
This is based on joint work in progress with Theo Johnson-Freyd.