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

Koytcheff, Robin¹

¹Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1467-1501

Résumé

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.

Détail

DOI : 10.2140/agt.2009.9.1467

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

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