Volume 9, issue 3 (2009)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

Robin Koytcheff

Algebraic & Geometric Topology 9 (2009) 1467–1501
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.

Keywords
knot spaces, configuration spaces, integration along the fiber, Pontrjagin–Thom construction
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 55R12, 55R80
References
Publication
Received: 3 December 2008
Revised: 26 June 2009
Accepted: 30 June 2009
Published: 29 July 2009
Authors
Robin Koytcheff
Department of Mathematics
Stanford University
Stanford, CA 94305
USA