Algebraic & Geometric Topology

Configuration spaces and Vassiliev classes in any dimension

Alberto S Cattaneo, Paolo Cotta-Ramusino, and Riccardo Longoni

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The real cohomology of the space of imbeddings of S1 into 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.

Article information

Algebr. Geom. Topol., Volume 2, Number 2 (2002), 949-1000.

Received: 2 August 2002
Accepted: 12 October 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58D10: Spaces of imbeddings and immersions
Secondary: 55R80: Discriminantal varieties, configuration spaces 81Q30: Feynman integrals and graphs; applications of algebraic topology and algebraic geometry [See also 14D05, 32S40]

configuration spaces Vassiliev invariants de Rham cohomology of spaces of imbeddings immersions Chen's iterated integrals graph cohomology


Cattaneo, Alberto S; Cotta-Ramusino, Paolo; Longoni, Riccardo. Configuration spaces and Vassiliev classes in any dimension. Algebr. Geom. Topol. 2 (2002), no. 2, 949--1000. doi:10.2140/agt.2002.2.949.

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