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18
Operads and knot spaces
 J. Amer. Math. Soc
"... Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent ..."
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Cited by 24 (2 self)
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Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated
New perspectives in self linking
 Adv. Math
"... Abstract. 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 npoint configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth ma ..."
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Cited by 10 (6 self)
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Abstract. 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 npoint 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 integervalued 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 finitetype 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. Contents
A HAEFLIGER STYLE DESCRIPTION OF THE EMBEDDING CALCULUS TOWER
"... Abstract. Let M and N be smooth manifolds. The calculus of embeddings produces, for every k ≥ 1, a best degree ≤ k polynomial approximation to the cofunctor taking an open V ⊂ M to the space of embeddings from V to N. In this paper a description of these polynomial approximations in terms of equivar ..."
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Cited by 7 (1 self)
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Abstract. Let M and N be smooth manifolds. The calculus of embeddings produces, for every k ≥ 1, a best degree ≤ k polynomial approximation to the cofunctor taking an open V ⊂ M to the space of embeddings from V to N. In this paper a description of these polynomial approximations in terms of equivariant mapping spaces is given, for k ≥ 2. The description is new only for k ≥ 3. In the case k = 2 we recover Haefliger’s approximation and the known result that it is the best degree ≤ 2 approximation. Let M and N be smooth manifolds, without boundary for simplicity, dim(M) = m and dim(N) = n where n> 3. The calculus of embeddings [10], [11], [3], [2] produces certain ‘Taylor ’ approximations Tkemb(M, N) to the space emb(M, N) of smooth embeddings from M to N. In more detail, there are maps
Configuration space integrals and Taylor towers for spaces of knots. Submitted
, 2004
"... 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 integ ..."
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Cited by 5 (0 self)
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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 BottTaubes 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 wellknown result that BottTaubes integrals combine to yield a universal finite type knot invariant. 1.
Coformality and the rational homotopy groups of spaces of long knots
, 2007
"... Abstract. We show that the BousfieldKan spectral sequence which computes the rational homotopy groups of the space of long knots in R d for d ≥ 4 collapses at the E 2 page. The main ingredients in the proof are Sinha’s cosimplicial model for the space of long knots and a coformality result for the ..."
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Cited by 5 (4 self)
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Abstract. We show that the BousfieldKan spectral sequence which computes the rational homotopy groups of the space of long knots in R d for d ≥ 4 collapses at the E 2 page. The main ingredients in the proof are Sinha’s cosimplicial model for the space of long knots and a coformality result for the little balls operad. 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).
MULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS
"... 3. Getting the ambient space to be connected 6 4. Theorems 1 and 2 are equivalent when r ≥ 1. 8 ..."
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Cited by 3 (1 self)
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3. Getting the ambient space to be connected 6 4. Theorems 1 and 2 are equivalent when r ≥ 1. 8
Embeddings in the 3/4 range
, 2003
"... We prove a theorem concerning the obstructions to turning an immersion f: M m → R n into an embedding in the range 3n ≥ 4m + 5. It is a secondary obstruction, and exists only when a certain primary obstruction, due to André Haefliger, vanishes. Our theorem 4 states that the obstruction lives in a ce ..."
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We prove a theorem concerning the obstructions to turning an immersion f: M m → R n into an embedding in the range 3n ≥ 4m + 5. It is a secondary obstruction, and exists only when a certain primary obstruction, due to André Haefliger, vanishes. Our theorem 4 states that the obstruction lives in a certain twisted cobordism group, and its vanishing implies the existence of an embedding in the regular homotopy class of f in the range indicated. We build on the work Haefliger, who measured the obstruction in the range 2n ≥ 3m + 3, and whose work generalizes the Whitney trick. We use Tom Goodwillie’s calculus of functors, following Michael Weiss, to help us organize and prove our result.
Contents
, 2003
"... Abstract. 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 elsew ..."
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Abstract. 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 E 1 terms which converge to their cohomology and homotopy groups.
MULTIPLE POINTS OF IMMERSIONS
, 2002
"... Abstract. Given smooth manifolds V n and M m, an integer k, and an immersion f: V � M, we have constructed an obstruction for existence of regular homotopy of f to an immersion f ′ : V � M without kfold points. This obstruction takes values in certain framed bordism group, and for (k + 1)(n + 1)≤km ..."
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Abstract. Given smooth manifolds V n and M m, an integer k, and an immersion f: V � M, we have constructed an obstruction for existence of regular homotopy of f to an immersion f ′ : V � M without kfold points. This obstruction takes values in certain framed bordism group, and for (k + 1)(n + 1)≤km turns out to be complete. 1.