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The rational homology of the space of long knots in codimension greater than two, preprint
"... 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 ho ..."
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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 ddisks operad and of Sinha’s cosimplicial model for the space of long knots arising from GoodwillieWeiss 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 BousfieldKan spectral sequences of a truncated cosimplicial space. 1.
CALCULUS OF FUNCTORS, OPERAD FORMALITY, AND RATIONAL HOMOLOGY OF EMBEDDING SPACES
, 2007
"... Abstract. Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V) be the homotopy fiber of the map Emb(M, V) − → Imm(M, V). This paper is about the rational homology of Emb(M, V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M, V) ↦ → HQ ∧ Emb(M, V) ..."
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Cited by 4 (3 self)
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Abstract. Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V) be the homotopy fiber of the map Emb(M, V) − → Imm(M, V). This paper is about the rational homology of Emb(M, V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M, V) ↦ → HQ ∧ Emb(M, V)+. Our main theorem states that if dim V ≥ 2ED(M) + 1 (where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E 1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich’s theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor HQ ∧ Emb(M, V)+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of Emb(−, V).
HOMOTOPY GRAPHCOMPLEX FOR CONFIGURATION AND KNOT SPACES
, 2006
"... Abstract. In the paper we prove that the primitive part of the Sinha homology spectral sequence E 2term for the space of long knots is rationally isomorphic to the homotopy E 2term. We also define natural graphcomplexes computing the rational homotopy of configuration and of knot spaces. 1. ..."
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Cited by 3 (0 self)
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Abstract. In the paper we prove that the primitive part of the Sinha homology spectral sequence E 2term for the space of long knots is rationally isomorphic to the homotopy E 2term. We also define natural graphcomplexes computing the rational homotopy of configuration and of knot spaces. 1.
HODGE DECOMPOSITION IN THE HOMOLOGY OF LONG KNOTS
, 812
"... Abstract. The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function fo ..."
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Abstract. The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graphcomplexes we show that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. BarNatan. In the Appendix we present tables of computer calculations of the Euler
The topology of spaces of knots
, 2003
"... We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of mapping spaces and another which is cosimplicial. At the geometric heart of these constructions is the evaluation map, used elsewhere for e ..."
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We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of mapping spaces and another which is cosimplicial. At the geometric heart of these constructions is the evaluation map, used elsewhere for example to define linking number and BottTaubes integrals. Our models are weakly homotopy equivalent to the corresponding knot spaces when the dimension of the ambient manifold is greater than three. There are spectral sequences with identifiable E¹ terms which converge to their cohomology and homotopy groups.