Topology and Geometric Group Theory Seminar

Yuri BerestCornell University
Contact homology of spaces

Tuesday, March 10, 2020 - 1:30pm
Malott 203

Knot contact homology is an interesting geometric invariant of a knot K in R^3 defined by Floer-​theoretic counting of pseudo-holomorphic disks in the sphere conormal bundle of K. This invariant -- introduced by L. Ng in 2008 (based on the earlier work of Eliashberg, Givental, Hofer and others) -- has been extensively studied in recent years by means of symplectic geometry and topology. In this talk, I will explain an algebraic construction of knot contact homology based on `abstract' homotopy theory of (small) DG categories. This construction reveals a close analogy with representation homology, i.e. derived representation varieties of fundamental groups. The latter are defined for arbitrary spaces (not only for knot complements), which raises naturally the question about a possible generalization of knot contact homology. I will sketch such a generalization using homotopy theory of cubical sets as combinatorial models of spaces.