Configuration spaces and Vassiliev classes in any dimension

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.

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Keywords

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

Mathematical Subject Classification 2000

Primary: 58D10

Secondary: 55R80, 81Q30

References

Forward citations

Publication

Received: 2 August 2002

Accepted: 12 October 2002

Published: 25 October 2002

Authors

Alberto S Cattaneo
Mathematisches Institut Universität Zürich–Irchel Winterthurerstrasse 190 CH-8057 Zürich Switzerland

Paolo Cotta-Ramusino
Dipartimento di Fisica Università degli Studi di Milano & INFN Sezione di Milano Via Celoria, 16 I-20133 Milano Italy

Riccardo Longoni
Dipartimento di Matematica “G. Castelnuovo” Università di Roma “La Sapienza” Piazzale Aldo Moro, 5 I-00185 Roma Italy

Volume 2, issue 2 (2002)

Alberto S Cattaneo, Paolo Cotta-Ramusino and Riccardo Longoni

Abstract