Lipschitz Homotopy Groups of the Heisenberg Groups

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ${\mathbb{H}^n}$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj?asz-Lukyanenko-Tyson constructed horizontal maps from S ^k to ${\mathbb{H}^n}$ which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map ${S^k \to \mathbb{H}^1}$ factors through a tree and is thus Lipschitz null-homotopic if ${k \geq 2}$ .     

球面稳定同伦群中的部分第三周期性元素族

考察了对于素数P≥7,当1≤r＜p-3/2时,环谱Vr(2)的交换性、结合性以及其它一些性质,并且还得出了球面稳定同伦群的部分第三周期性元素族γtpn/r(n≥1,t≥1,1≤r≤2n相似文献   

Homotopy Spheres in Formal Language

The set of nonempty proper subwords of a word is either contractible or a homotopy n-sphere. There is a simple algorithm which computes n. The existence of spherical words is investigated, and the words which yield spheres are determined. A language which is closed under subwords has a finite number of components, and each component has a finitely generated fundamental group. For each n greater than 1, there is a language on two letters which has the homotopy type of an infinite cluster of n-sphere. There is a language on two letters which has nontrivial homology in each dimension greater than 1. If a language is closed under subwords and has bounded period, then it has the homotopy type of a finite polyhedron.     

A NEW FAMILY OF FILTRATION S ＋ 5 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

In this article, by the algebraic method, the author proves the existence of a new nontrivial family of filtration s ＋ 5 in the stable homotopy groups of spheres πrS,which is represented by 0 ≠γ^-s＋3hnhm∈Ext^s＋5,A ^t（Zp,Zp）in the Adams spectral sequence,where r=q（p^m＋p^n＋（s＋3）p^2＋（s＋2）p＋（s＋1））-5,t=p^mq＋p^nq＋（s＋3）p^2q＋（s＋2）pq＋（s＋1）q＋s,p≥7,m≥n＋2〉5,0≤s〈p-3,q=2（p-1）.     

Rectifiability and Lipschitz extensions into the Heisenberg group

Denote by mathbbHⁿ{mathbb{H}^n} the 2n + 1 dimensional Heisenberg group. We show that the pairs (mathbbR^k ,mathbbHⁿ){(mathbb{R}^k ,mathbb{H}^n)} and (mathbbH^k ,mathbbHⁿ){(mathbb{H}^k ,mathbb{H}^n)} do not have the Lipschitz extension property for k  >  n.     

New Families in the Stable Homotopy of Spheres Revisited

This paper constructs a new family in the stable homotopy of spheres π _{t −6} S represented by h _n g ₀γ₃∈E ^{6 t} ₂ in the Adams spectral sequence which revisits the b _{n −1} g ₀γ₃-elements ∈π _{t −7} S constructed in [3], where t = 2p ⁿ (p− 1) + 6(p ² + p + 1)(p− 1) and p≥ 7 is a prime, n≥ 4. Received October 7, 1998, Revised May 8, 2000, Accepted August 8, 2000.     

Geodesics in Heisenberg Groups

We obtain equations of geodesic lines in Heisenberg groups H²ⁿ⁺¹and prove that the ideal boundary of the Heisenberg group H²ⁿ⁺¹is a sphere S^2n-1with a natural CR-structure and corresponding Carnot-Carathéodory metric, i.e. it is a one-point compactification of the Heisenberg group H^2n-1of the next dimension in a row.     

Homotopy Groups of the Combinatorial Grassmannian

We prove that the homotopy groups of the oriented matroid Grassmannian MacP(k,n) are stable as n , that , and that there is a surjection . Received November 4, 1996, and in revised form April 22, 1997.     

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.     

Lipschitz continuity of refinable functions on the Heisenberg group

In this paper, the Lipschitz continuity of refinable functions related to the general acceptable dilations on the Heisenberg group will be investigated in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces related to a kind of special acceptable dilations.     

Some Remarks About Heisenberg Groups

In this paper, we give two inequalities and another characterizations about Heisenberg groups which do not coincide with the related results in classical case.     

Generalizations of Fox Homotopy Groups,Whitehead Products,and Gottlieb Groups

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [Σ(V×W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 320–328, March, 2005.     

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.     

Generalized Lipschitz Spaces on Vilenkin Groups

It is shown that, for certain choices of the defining indices, the generalized Lipschitz spaces on Vilenkin groups are included in certain Figà-Talamanca spaces A_p, and that the Fourier series of functions in the latter spaces converge uniformly. This result includes an extension of the classical Dini test, and endpoint versions of Bernstein's theorem on absolute convergence of Fourier series.     

Dynamic Random Walks on Heisenberg Groups

We prove a Guivarc’h law of large numbers and a central limit theorem for dynamic random walks on Heisenberg groups. The limiting distribution is explicitely given. To our knowledge this is the first study of dynamic random walks on non-commutative Lie groups.      

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the $L^{\infty }$ -norm of its intrinsic gradient.     

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.     

Pompeiu Problem on Product of Heisenberg Groups

In this paper, we address the Pompeiu problem for a product of Heisenberg groups. We consider this problem both for cases of a ball and for a bidisk. Furthermore, we address this problem for a product of the Heisenberg group with Euclidean space.     

Riesz Transforms on Groups of Heisenberg Type

The aim of this article is to prove the following theorem. Theorem Let p be in (1,∞), ℍ _n,m a group of Heisenberg type, ℛ the vector of the Riesz transforms on ℍ _n,m . There exists a constant C _p independent of n and m such that for every f∈L ^p (ℍ _n,m )