A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

Abstract

We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by Budney and Cohen. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots.