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2009 A homotopy-theoretic view of Bott–Taubes integrals and knot spaces
Robin Koytcheff
Algebr. Geom. Topol. 9(3): 1467-1501 (2009). DOI: 10.2140/agt.2009.9.1467


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.


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Robin Koytcheff. "A homotopy-theoretic view of Bott–Taubes integrals and knot spaces." Algebr. Geom. Topol. 9 (3) 1467 - 1501, 2009.


Received: 3 December 2008; Revised: 26 June 2009; Accepted: 30 June 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1175.57012
MathSciNet: MR2530124
Digital Object Identifier: 10.2140/agt.2009.9.1467

Primary: 57M27
Secondary: 55R12, 55R80

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.9 • No. 3 • 2009
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