Configuration spaces and Vassiliev classes in any dimension

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.