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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


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.

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
Forward citations
Received: 2 August 2002
Accepted: 12 October 2002
Published: 25 October 2002
Alberto S Cattaneo
Mathematisches Institut
Universität Zürich–Irchel
Winterthurerstrasse 190
CH-8057 Zürich
Paolo Cotta-Ramusino
Dipartimento di Fisica
Università degli Studi di Milano & INFN Sezione di Milano
Via Celoria, 16
I-20133 Milano
Riccardo Longoni
Dipartimento di Matematica “G. Castelnuovo”
Università di Roma “La Sapienza”
Piazzale Aldo Moro, 5
I-00185 Roma