#### Volume 2, issue 2 (2002)

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Configuration spaces and Vassiliev classes in any dimension

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

Algebraic & Geometric Topology 2 (2002) 949–1000
 arXiv: math.GT/9910139
##### 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, immersions, Chen's iterated integrals, graph cohomology
##### Mathematical Subject Classification 2000
Primary: 58D10
Secondary: 55R80, 81Q30
##### 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