On nontriviality of certain homotopy groups of spheres

Ivanov, Sergei O.; Mikhailov, Roman; Wu, Jie

doi:10.4310/hha.2016.v18.n2.a18

Cited by 7 publications

(4 citation statements)

References 18 publications

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“…In the case of loops the π 3 (S 2 )-factor in the exact sequence (18) is not in the image of π 1 (i F ) and in higher homotopy groups the π k+2 (S 2 ) ∼ = π k (FLeg N,jN (S 3 , ξ std ))-factor is not in the image of π k (i F ). These groups are always non-trivial [IMW16].…”

Section: A Decomposition Of the Space Of Formal Legendrian Embeddings...mentioning

confidence: 99%

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

Fernández¹,

Martínez-Aguinaga²,

Presas³

2020

Preprint

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We prove that that the homotopy type of the path connected component of the identity in the contactomorphism group is characterized by the homotopy type of the diffeomorphism group plus some data provided by the topology of the formal contactomorphism space. As a consequence, we show that every connected component of the space of Legendrian long knots in R 3 has the homotopy type of the corresponding smooth long knot space. This implies that any connected component of the space of Legendrian embeddings in S 3 is homotopy equivalent to the space K(G, 1) × U(2), with G computed by A. Hatcher and R. Budney. Similar statements are proven for Legendrian embeddings in R 3 and for transverse embeddings in S 3 . Finally, we compute the homotopy type of the contactomorphisms of several tight 3-folds: S 1 × S 2 , Legendrian fibrations over compact orientable surfaces and finite quotients of the standard 3-sphere. In fact, the computations show that the method works whenever we have knowledge of the topology of the diffeomorphism group. We prove several statements on the way that have interest by themselves: the computation of the homotopy groups of the space of non-parametrized Legendrians, a multiparametric convex surface theory, a characterization of formal Legendrian simplicity in terms of the space of tight contact structures on the complement of a Legendrian, the existence of common trivializations for multi-parametric families of tight R 3 , etc. Contents 1. Introduction. 1.1. Outline. 1.2. Acknowledgements. 2. Preliminaries. 2.1. Convex surface theory in tight contact 3-manifolds. 2.2. Fibrations in contact topology. 2.3. The space of Darboux balls in a contact manifold. 2.4. Legendrian embeddings in (S 3 , ξ std ). 2.5. Smooth Case. Hatcher's Theorems.

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Section: A Decomposition Of the Space Of Formal Legendrian Embeddings...mentioning

confidence: 99%

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

Fernández¹,

Martínez-Aguinaga²,

Presas³

2020

Preprint

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“…. all homotopy groups in degree > 1 of S 2 are non-zero (see [19]). In general, the sequence of finite abelian groups π n (S 2 ), n ≥ 4, is one of the most mysterious objects in math, it is difficult to speculate how far we are from its understanding.…”

Section: Wu-type Formulasmentioning

confidence: 99%

Homotopy patterns in group theory

Mikhailov¹

2021

Preprint

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This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of non-additive functors, homotopy groups of spheres, group homology etc appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.

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“…It is natural to wonder whether finite subdivision suffices to recover the correct homotopy type. We do not address this question here, noting only that the method of proof completely breaks down; indeed, the homotopy groups of Conf 2 (D 3 ) ≃ S 2 are all non-zero above degree one [IMW16].…”

Section: Configuration Complexesmentioning

confidence: 99%

Subdivisional Spaces and Graph Braid Groups

Drummond-Cole

Knudsen

2019

Doc. Math.

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We study the problem of computing the homology of the configuration spaces of a finite cell complex X. We proceed by viewing X, together with its subdivisions, as a subdivisional space-a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose X and show that the homology of the configuration spaces of X is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology.Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to Świątkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.

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On nontriviality of certain homotopy groups of spheres

Abstract: For n ≥ 2, the homotopy groups πn(S 2 ) are non-zero.

Cited by 7 publications

References 18 publications

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

Homotopy patterns in group theory

Subdivisional Spaces and Graph Braid Groups

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