Configuration spaces and Vassiliev classes in any dimension

Abstract

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.html

References in corpus (7)

• Deformation quantization of Poisson manifolds, I
• Embeddings from the point of view of immersion theory: Part I
• Integral Invariants of 3-Manifolds. II
• Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops
• Topological Field Theory Interpretation of String Topology
• Loop and Path Spaces and Four-Dimensional BF Theories: Connections, Holonomies and Observables
• Algebraic structures on graph cohomology

Cited by in corpus (20)

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• Non-trivalent graph cocycle and cohomology of the long knot space
• Algebraic structures on graph cohomology
• Poisson structures on the homology of the space of knots
• Configuration space integrals for embedding spaces and the Haefliger invariant
• A homotopy-theoretic view of Bott-Taubes integrals and knot spaces
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3
• 1-loop graphs and configuration space integral for embedding spaces
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• The Milnor triple-linking number of string links by cut-and-paste topology
• A Kontsevich integral of order 1
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• Vassiliev Invariants for Flows Via Chern-Simons Perturbation Theory
• The Fox-Hatcher cycle and a Vassiliev invariant of order three
• Graphing, homotopy groups of spheres, and spaces of long links and knots