Caltech/USC/UCLA Joint Topology Seminar
USC KAP 414
Roughly, factorization homology pairs an n-category and an n-manifold to produce a chain complex. Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (i.e. independent of auxiliary choices), continuous (i.e. carries a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.
Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models. For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincaré duality, which yields some surprising dualities among TQFTs. A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.
The first part of this talk will tour through some essential definitions in what's described above. The second part of the talk will focus on familiar instances of factorization homology, highlighting the Poincaré/Koszul duality result. The last part of the talk will speculate on more such instances.