Algebraic and Geometric Topology 2 (2002), paper no. 39, pages 949-1000.

Configuration spaces and Vassiliev classes in any dimension

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

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.

Keywords. Configuration spaces, Vassiliev invariants, de Rham cohomology of spaces of imbeddings and immersions, Chen's iterated integrals, graph cohomology

AMS subject classification. Primary: 58D10. Secondary: 55R80, 81Q30.

DOI: 10.2140/agt.2002.2.949

E-print: arXiv:math.GT/9910139

Submitted: 2 August 2002. Accepted: 12 October 2002. Published: 25 October 2002.

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

Mathematisches Institut, Universitat Zurich-Irchel, Winterthurerstrasse 190
CH-8057 Zurich, Switzerland

Dipartimento di Fisica, Universita degli Studi di Milano & INFN Sezione di Milano
Via Celoria, 16, I-20133 Milano, Italy

Dipartimento di Matematica `G. Castelnuovo', Universita di Roma, La Sapienza
Piazzale Aldo Moro, 5, I-00185 Roma, Italy

Email: asc@math.unizh.ch, cotta@mi.infn.it,longoni@mat.uniroma1.it

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