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 [J. Math. Phys. 35 (1994) 5247-5287], 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. Inspired by results of Budney and Cohen [Geom. Topol. 13 (2009) 99-139], we study how this integration is compatible with homology operations on the space of long knots. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots.