Multiple disjunction for spaces of smooth embeddings

Goodwillie, Thomas G.; Klein, John R.

doi:10.1112/jtopol/jtv008

Cited by 31 publications

(37 citation statements)

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“…Essentially because the classical adjunction can be factorized as follows: Seq(S) Seq(S) P 0 Bimod P-Q . In this section, we give a presentation of the left adjoint functors (12). Thereafter, we adapt the Boardman-Vogt resolution in the context of (k-truncated) bimodules to obtain functorial cofibrant replacements.…”

Section: Remark 23mentioning

confidence: 99%

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From maps between coloured operads to Swiss-Cheese algebras

Ducoulombier¹

2018

Ann. inst. Fourier

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In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad SC 1 . Thereafter, we show that the pair formed by the space of long embeddings and the manifold calculus limit of (l)-immersions from R d to R n is an SC d+1 -algebra assuming the Dwyer-Hess' conjecture.are proved to be weakly equivalent to explicit SC d+1 -algebras.Organization of the paper. The paper is divided into 4 sections. The first section gives an introduction on coloured operads and (infinitesimal) bimodules over coloured operads as well as the truncated versions of these notions. In particular, the little cubes operad, the Swiss-Cheese operad and the non-(l)-overlapping little cubes bimodule are defined.In the second section, we give a presentation of the left adjoint functor to the forgetful functor from the category of (P-Q) bimodules to the category of S-sequences. This presentation is used to endow Bimod P-Q with a cofibrantly generated model category structure. Thereafter, we prove that a Boardman-Vogt type resolution yields explicit and functorial cofibrant replacements in the model category of (P-Q) bimodules. We also show that similar statements hold true for truncated bimodules.The third section is devoted to the proof of the main theorem 3.20. For this purpose, we give a presentation of the functor L and prove that the adjunction (4) is a Quillen adjunction. Then we change slightly the Boardman-Vogt resolution introduced in Section 2 in order to obtain explicit cofibrant replacements in the category Op[O ; ∅]. Finally, by using Theorem 3.20 and the Dwyer-Hess' conjecture, we identify SC d+1algebras from maps of operads η : CC d → O .In the last section we give an application of our results to the space of long embeddings in higher dimension. We introduce quickly the Goodwillie calculus as well as the relation between the manifold calculus tower and the mapping space of infinitesimal bimodules. Then we show that the pairs (5) are weakly equivalent to explicit typical SC d+1 -algebras.Convention. By a space we mean a compactly generated Hausdorff space and by abuse of notation we denote by Top this category (see e.g. [18, section 2.4]). If X, Y and Z are spaces, then Top(X; Y) is equipped with the compact-open topology in order to have a homeomorphism Top(X; Top(Y; Z)) Top(X × Y; Z). By using the Serre fibrations, the category Top is endowed with a cofibrantly generated monoidal model structure. In the paper the categories considered are enriched over Top.By convention C ∞ d (0) is the one point topological space and the operadic composition • i with this point consists in forgetting the i-th little cube. Definition 1.5. Let S be a set and O be an S-operad. An algebra over the operad O, or O-algebra, is given by a family of topological spaces X := {X s } s∈S endowed with operations µ : O(s 1 , . . . , s n ; s n+1 ) × X s 1 × · · · × X sn −→ X s n+1 , compatible with the operadic composit...

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Section: Remark 23mentioning

confidence: 99%

“…[12] Let M and N be two smooth manifolds of dimension d and n respectively. If n − d − 2 > 0, then the Taylor towers associated to the functors Emb(− ; N), Imm(− ; N) and Emb(− ; N) converge.If one considers the embedding spaces and immersion spaces with compact support, then one has to change slightly the construction of the Taylor towers.…”

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confidence: 99%

From maps between coloured operads to Swiss-Cheese algebras

Ducoulombier¹

2018

Ann. inst. Fourier

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“…Remark 3.1. The version of the Poincaré embedding space appearing here is slightly different from the one in [GK1,defn. 2.8].…”

Section: Poincaré Embeddingsmentioning

confidence: 99%

“…Our definition here amounts to taking a certain open and closed subspace of I (A) rather than a homotopy fiber. This definition is equivalent to the one in [GK1] because the component of D n ∈ I (S n−1 ) is contractible.…”

Section: Poincaré Embeddingsmentioning

confidence: 99%

“…Let E sm (P, N) be the space of smooth (C ∞ ) embeddings from P into the interior of N. The manifold calculus of Goodwillie and Weiss produces a tower of fibrations · · · → E sm 2 (P, N) → E sm 1 (P, N) and compatible maps E sm (P, N) → E sm j (P, N). If we assume that P admits a handle decomposition with handles of index at most n − 3, then the maps E sm (P, N) → E sm j (P, N) have connectivity given by a linear function of j with positive slope, so in this case the tower strongly converges [GW], [GK2]. Furthermore, E sm 1 (P, N) has the homotopy type of the space of smooth immersions from P to N. For j ≥ 2, the layers of the tower, i.e., the homotopy fibers of the maps E sm j (P, N) → E sm j−1 (P, N), have an explicit description in terms of configuration spaces.…”

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Embeddings, Normal Invariants and Functor Calculus

Klein

2016

Nagoya Math. J.

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Abstract. This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of "unlinked" Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie's homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

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Embedding Calculus and the Little Discs Operads

Turchin

2018

MATRIX Book Series

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Multiple disjunction for spaces of smooth embeddings

Abstract: Abstract. We obtain multirelative connectivity statements about spaces of smooth embeddings, deducing these from a similar result about spaces of Poincaré embeddings that was established in [GK1] and a similar result about condordance embeddings that was established in [G1].

Cited by 31 publications

References 15 publications

From maps between coloured operads to Swiss-Cheese algebras

From maps between coloured operads to Swiss-Cheese algebras

Embeddings, Normal Invariants and Functor Calculus

Embedding Calculus and the Little Discs Operads

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