Spaces of smooth embeddings, disjunction and surgery

Goodwillie, Thomas G.; Klein, John R.; Weiss, Michael

doi:10.1515/9781400865215-007

Cited by 33 publications

(48 citation statements)

References 56 publications

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“…We will only need a much weaker convergence result, whose proof is accordingly easier. The "weak convergence theorem" says that the above Taylor tower converges if 2 dim M +2<dim N and a proof can be found in the remark after Corollary 4.2.4 in [10]. The weak convergence result also holds for HQ∧Emb(M, N ) + by the main result of [24].…”

Section: Introductionmentioning

confidence: 90%

Calculus of functors, operad formality, and rational homology of embedding spaces

2007

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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 bi-functor (M, V ) → HQ ∧ Emb(M, V )+. Our main theorem states that if dim V ≥ 2 ED(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 functorThe 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 ).

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Section: Introductionmentioning

confidence: 90%

Calculus of functors, operad formality, and rational homology of embedding spaces

2007

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“…In particular, it is suggested in [15,Example 5.1.4] that this approach should also give a spectral sequence for computing the homology of spaces of long knots. Indeed, it later turned out that it does, and that this spectral sequence was equivalent to Vassiliev's.…”

Section: Introductionmentioning

confidence: 99%

The rational homology of spaces of long knots in codimension>2

2010

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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 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.

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“…introduced by the fourth author [36] building on the calculus of embeddings of Goodwillie et al [15][16][17]41,42]. When the dimension of M is greater than three, the map C n induces isomorphisms on homology and homotopy groups up to degree nðdimðMÞ À 3Þ: For three-manifolds we conjecture a strong relation to finite-type invariants.…”

Section: Introductionmentioning

confidence: 83%

“…16. The map C n : EmbðI; MÞ-AM n ðMÞ is a model for the nth degree approximation to EmbðI; MÞ in the calculus of embeddings.…”

Section: The Mapping Space Modelmentioning

confidence: 99%

New perspectives on self-linking

Budney

Conant

Scannell

et al. 2005

Advances in Mathematics

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We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. r

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Spaces of smooth embeddings, disjunction and surgery

Cited by 33 publications

References 56 publications

Calculus of functors, operad formality, and rational homology of embedding spaces

Calculus of functors, operad formality, and rational homology of embedding spaces

The rational homology of spaces of long knots in codimension>2

New perspectives on self-linking

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