Algebraic & Geometric Topology

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

Robin Koytcheff

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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.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1467-1501.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797036

Digital Object Identifier
doi:10.2140/agt.2009.9.1467

Mathematical Reviews number (MathSciNet)
MR2530124

Zentralblatt MATH identifier
1175.57012

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 55R12: Transfer 55R80: Discriminantal varieties, configuration spaces

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

Citation

Koytcheff, Robin. A homotopy-theoretic view of Bott–Taubes integrals and knot spaces. Algebr. Geom. Topol. 9 (2009), no. 3, 1467--1501. doi:10.2140/agt.2009.9.1467. https://projecteuclid.org/euclid.agt/1513797036


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