We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of R m into R n . We view the space of embeddings as the value of a certain functor at R m , and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when 2m + 1 < n, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.is a weak homotopy equivalence if m + 3 ≤ n. Combining this with Theorem 0.1, we conclude that the following natural map is an equivalence when m + 3 ≤ nThe case m = 1 of Theorem 0.1 is related to Sinha's cosimplicial model for the space of long knots [23]. Indeed, the homotopy totalization of Sinha's cosimplicial space can be
We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on different ways how the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.
We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad. P A 6.1. Poincaré Lemma for Ω * P A 6.2. Sheaf propery of Ω * P A 6.3. Extendability of the simplicial set Ω P A (∆ • ) 6.4. The weak equivalence Ω * P A ≃ A P L 7. Monoidal equivalences 8. Oriented semi-algebraic bundles and integration along the fiber 8.1. Properties of SA bundles 8.2. Properties of integration along the fiber
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 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 Bousfield-Kan spectral sequences of a truncated cosimplicial space.
In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-koverlapping discs. At the end we briefly explain how this algebraic structure intervenes in the study of spaces of non-k-equal immersions.
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