Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.

ABSTRACT. This paper analyzes in details the Batalin–Vilkovisky quantization procedure for BF theories on n-dimensional manifolds and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF theories with additional polynomial B-interactions are discussed in any dimensions. The paper also contains the explicit proofs to the Theorems stated in [16].

Abstract. 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. 1.

ABSTRACT. A generalization of Wilson loop observables for BF theories in any dimension is introduced in the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the paper has only trivalent interactions and, irrespective of the actual dimension, looks like a 3-dimensional Chern–Simons theory. 1.

Abstract. We determine the rational homology of the space of long knots in R d for d ≥ 4. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E 1 page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with a bracket of degree d −1, which can be obtained as the homology of an explicit graph complex and is in theory completely computable. Our proof is a combination of a relative version of Kontsevich’s formality of the little d-disks operad and of Sinha’s cosimplicial model for the space of long knots arising from Goodwillie-Weiss embedding calculus. As another ingredient in our proof, we introduce a generalization of a construction that associates a cosimplicial object to a multiplicative operad. Along the way we also establish some results about the Bousfield-Kan spectral sequences of a truncated cosimplicial space. 1.

Abstract. We describe Taylor towers for spaces of knots arising from Goodwillie’s calculus of functors and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We prove the vanishing results in detail to show that certain combinations of integrals, dictated by trivalent diagrams, yield cohomology classes of the stages of the tower, just as they do for ordinary knots. We then use this factorization of Bott-Taubes integrals through the Taylor tower to deduce a vanishing result for a spectral sequence converging to the cohomology of spaces of knots. We also give another proof of the well-known result that Bott-Taubes integrals combine to yield a universal finite type knot invariant. 1.

Abstract. In the paper we prove that the primitive part of the Sinha homology spectral sequence E 2-term for the space of long knots is rationally isomorphic to the homotopy E 2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces. 1.

Abstract not found

Abstract. Simplicial and ∆-structures of configuration spaces are investigated. New connections between the homotopy groups of the 2-sphere and the braid groups are given. The higher homotopy groups of the 2- sphere are shown to be derived groups of the braid groups over the 2-sphere. Moreover the higher homotopy groups of the 2-sphere are shown to be isomorphic to the

ABSTRACT. Kontsevich constructed universal characteristic classes of smooth bundles with fiber a framed homology sphere, which is known in the 3-dimensional case to be a universal finite type invariant. The purpose of the present paper is twofold. First, we obtain a bordism invariant of smooth unframed bundles with fiber a 5-dimensional homology sphere as a sum of the simplest Kontsevich class and the second signature defect. Second, we introduce the notion of clasper bundles. By using clasper bundles, we show that Kontsevich’s universal characteristic classes are highly non-trivial in the case of fiber dimension 7.