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