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.
Keywords
knot spaces, configuration spaces, integration along the
fiber, Pontrjagin–Thom construction
