Lipschitz Homotopy Groups of the Heisenberg Groups
Lipschitz Homotopy Groups of the Heisenberg Groups
复制标题
海森堡群的利普希茨同伦群
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Robert Young
中科院分区：
文献类型：
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作者：
S. Wenger;Robert Young
从n维空间到（2n + 1）维海森堡群的Lipschitz映射和水平映射是丰富的，而从高维空间到海森堡群的映射则受到更多的限制。DeJarnette-Hajamasz-Lukyanenko-Tyson构造了从Sk到$${mathbb{H}^n}$$Hn的水平映射，这些映射通过n-球面进行因子分解，并表明这些映射没有光滑的水平填充。然而，在本文中，我们建立在一个例子考夫曼，这些地图有时有Lipschitz填充。这表明空间的Lipschitz群和光滑水平同伦群可能不同。相反，我们证明了任何Lipschitz映射$${S^k o mathbb{H}^1}$$Sk→H1因子通过树，因此是Lipschitz零同伦的，如果$${k geq 2}$$k≥2。
Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group $${mathbb{H}^n}$$Hn are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from Sk to $${mathbb{H}^n}$$Hn 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 $${S^k o mathbb{H}^1}$$Sk→H1 factors through a tree and is thus Lipschitz null-homotopic if $${k geq 2}$$k≥2 .