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A homotopy-theoretic view of Bott-Taubes integrals and knot spaces
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. Inspired by results of Budney and Cohen, 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. We then adapt the construction to be compatible with tools coming from the Goodwillie-Weiss embedding calculus, in particular Sinha's cosimplicial model for the space of knots.
- "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. Inspired by results of Budney and Cohen, 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. We then adapt the construction to be compatible with tools coming from the Goodwillie-Weiss embedding calculus, in particular Sinha's cosimplicial model for the space of knots."@en
- "A homotopy-theoretic view of Bott-Taubes integrals and knot spaces"@en