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

TL;DR

This paper develops a homotopy-theoretic framework for Bott-Taubes integrals, constructing cohomology classes in knot spaces via fiber integration and demonstrating compatibility with knot homology operations.

Contribution

It introduces a homotopy-theoretic approach to Bott-Taubes integrals, producing integral cohomology classes and relating them to knot homology operations.

Findings

Constructs cohomology classes in knot spaces using a Pontrjagin-Thom construction.

Shows compatibility of this cohomology with connect-sum operations on knots.

Derives a product formula for cohomology evaluations on homology classes.

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.