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.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L-7,L-1 and L-7,L-2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products. (C) 2004 Elsevier Ltd. All rights reserved

1. Definitions 1.1. Graded vector spaces. By a Z-graded vector space (or simply, graded vector space) we mean a direct sum A = ⊕ i∈Z A i of vector spaces over a field k of characteristic zero. The A i are called the components of A of degree i and the degree of a homogeneous element a ∈ A is denoted by |a|. We also denote by A[n] the graded vector space with degree shifted by n, namely,The tensor product of two graded vector spaces A and B is again a graded vector space whose degree r component is given byThe symmetric and exterior algebra of a graded vector space A are defined respectively as S(A) = T (A)/I S and (A) = T (A)/I ∧ , where T (A) = ⊕ n≥0 A ⊗n is the tensor algebra of A and I S (resp. I ∧ ) is the two-sided ideal generated by elements of the form a ⊗ b − (−1) |a| |b| b ⊗ a (resp. a ⊗ b + (−1) |a| |b| b ⊗ a), with a and b homogeneous elements of A. The images of A ⊗n in S(A) and (A) are denoted by S n (A) and n (A) respectively. Notice that there is a canonical decalage isomorphism1.2. Graded algebras and graded Lie algebras. We say that A is a graded algebra (of degree zero) if A is a graded vector space endowed with a degree zero bilinear associative product · : A⊗A → A. A graded algebra is graded commutative if the product satisfies the conditionfor any two homogeneous elements a, b ∈ A of degree |a| and |b| respectively. A graded Lie algebra of degree n is a graded vector space A endowed with a graded Lie bracket on A[n]. Such a bracket can be seen as a degree −n Lie bracket on A, i.e., as bilinear operation {·, ·} : A ⊗ A → A[−n] satisfying graded antisymmetry and graded Jacobi relations: {a, b} = −(−1) (|a|+n)(|b|+n) {b, a} {a, {b, c}} = {{a, b}, c} + (−1) (|a|+n)(|b|+n) {a{b, c}}

Abstract. We construct nontrivial cohomology classes of the space Imb (S 1 , R n ) of imbeddings of the circle into R n , by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number n ≥ 4, a de Rham cohomology class on Imb (S 1 , R n ). We prove nontriviality of these classes by evaluation on the dual cycles.

We define algebraic structures on graph cohomology and prove that they correspond to algebraic structures on the cohomology of the spaces of imbeddings of S 1 or R into R n . As a corollary, we deduce the existence of an infinite number of nontrivial cohomology classes in Imb (S 1 , R n ) when n is even and greater than 3. Finally, we give a new interpretation of the anomaly term for the Vassiliev invariants in R 3 .1991 Mathematics Subject Classification. Primary 58D10; Secondary 81Q30. A. S. C. acknowledges partial support of SNF grant No. 20-100029/ 1. 1 Graph cohomologyWe briefly recall the definition of the graph complexes given in [5], Section 4. A graph consists of an oriented circle and many edges joining vertices. The vertices lying on the circle are called external vertices, those lying off the circle are called internal vertices and are required to be at least trivalent. We define the order k ≥ 0 of a graph to be minus its Euler characteristic, and the degree m ≥ 0 to be the deviation of the graph from being trivalent, namely:where e is the number of edges, v i the number of internal vertices and v e the number of external vertices of the graph.