Configuration spaces and Vassiliev classes in any dimension

Cattaneo, A S; Cotta-Ramusino, P; Longoni, R (2002). Configuration spaces and Vassiliev classes in any dimension. Algebraic & Geometric Topology, 2:949-1000 (electronic).

Abstract

The real cohomology of the space of imbeddings of
S1 into ℝn, 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.

Abstract

The real cohomology of the space of imbeddings of
S1 into ℝn, 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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