Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

Piotr Hajłasz; Armin Schikorra; Jeremy T. Tyson

doi:10.1007/s00039-014-0261-z

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

Piotr Hajłasz, Armin Schikorra, Jeremy T. Tyson

Mathematics

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Abstract

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, π_m^Lip(ℍ_n), in terms of properties of the classical homotopy group of the sphere, π_m(Sⁿ). As an application we provide a new simplified proof of the fact that (Formula presented), and we prove a new result that (Formula presented) for n = 1,2,. The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space W^1,p(M, ℍ_2n) when dim M ≥ 4n and 4n-1 ≤ p < 4n.

Original language	English (US)
Pages (from-to)	245-268
Number of pages	24
Journal	Geometric and Functional Analysis
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s00039-014-0261-z
State	Published - Feb 2014

Keywords

55Q25
55Q40
Primary 53C17
Secondary 46E35

ASJC Scopus subject areas

Analysis
Geometry and Topology

Online availability

10.1007/s00039-014-0261-z

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title = "Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups",

abstract = "We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, πmLip(ℍn), in terms of properties of the classical homotopy group of the sphere, πm(Sn). As an application we provide a new simplified proof of the fact that (Formula presented), and we prove a new result that (Formula presented) for n = 1,2,. The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space W1,p(M, ℍ2n) when dim M ≥ 4n and 4n-1 ≤ p < 4n.",

keywords = "55Q25, 55Q40, Primary 53C17, Secondary 46E35",

author = "Piotr Haj{\l}asz and Armin Schikorra and Tyson, {Jeremy T.}",

note = "Funding Information: Piotr Hajlasz was supported by NSF grant DMS-1161425. A.S. was supported by DAAD fellowship D/12/40670. J.T.T. was supported by NSF grant DMS-1201875.",

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doi = "10.1007/s00039-014-0261-z",

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volume = "24",

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AU - Hajłasz, Piotr

AU - Schikorra, Armin

AU - Tyson, Jeremy T.

N1 - Funding Information: Piotr Hajlasz was supported by NSF grant DMS-1161425. A.S. was supported by DAAD fellowship D/12/40670. J.T.T. was supported by NSF grant DMS-1201875.

PY - 2014/2

Y1 - 2014/2

N2 - We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, πmLip(ℍn), in terms of properties of the classical homotopy group of the sphere, πm(Sn). As an application we provide a new simplified proof of the fact that (Formula presented), and we prove a new result that (Formula presented) for n = 1,2,. The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space W1,p(M, ℍ2n) when dim M ≥ 4n and 4n-1 ≤ p < 4n.

AB - We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, πmLip(ℍn), in terms of properties of the classical homotopy group of the sphere, πm(Sn). As an application we provide a new simplified proof of the fact that (Formula presented), and we prove a new result that (Formula presented) for n = 1,2,. The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space W1,p(M, ℍ2n) when dim M ≥ 4n and 4n-1 ≤ p < 4n.

KW - 55Q25

KW - 55Q40

KW - Primary 53C17

KW - Secondary 46E35

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JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 1

ER -

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

Abstract

Keywords

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