A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps
A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps
复制标题
Lipschitz 地图对希尔伯特-史密斯猜想的证明
DOI：
10.1007/s002080050080
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发表时间：
1997
影响因子：
1.4
通讯作者：
Evgenij V. S˘c˘epin
中科院分区：
文献类型：
--
作者：
Dus˘an Repovs˘;Evgenij V. S˘c˘epin
经典的希尔伯特第五问题[14]询问是否每个（nite维）局部欧几里得拓扑群必然是李群。在实证中，冯·诺依曼[23]在1933年为紧群解决了这个问题，格里森[11]以及蒙哥马利和齐平[20]在1952年为局部紧群解决了这个问题。希尔伯特第五问题的一个更一般的版本，称为希尔伯特-史密斯猜想，断言在所有局部紧群中，只有李群G可以有效地作用在（有限维）流形M上（即每个g ∈ G\{e}移动M的至少一个点）[28]。从纽曼[24]和史密斯[29]的工作可以得出，当作用群G是p-adic整数群Ap时，这个猜想等价于它的特殊情况。1946年Bochner和蒙哥马利[3]证明了群G有效作用在流形M上的Hilbert-Smith猜想。斯科彭科夫和作者[25]使用光滑齐性的思想获得了一个更简单的几何证明：光滑流形M的紧致子集K M被称为光滑环境齐性，即对于每个x; y ∈ K，存在自同构h：（M;K; x）→（M;K; y）。证明了这一性质意味着K是M的光滑子流形（因此G = K是李群）。证明揭示了Rn的紧致子集的齐性和驯服理论之间的密切关系，这些子集被切球挤压（后一个问题在过去由不同的作者研究[6，10，12，16，17]）。另见哈恩的一篇非常有趣的论文[13]。希尔伯特-史密斯猜想的一个有趣的方法是通过Rn中具有强齐性性质的野生康托集。注意安托万的项链
The classical Hilbert 5th problem [14] asks whether every ( nite-dimensional) locally Euclidean topological group is necessarily a Lie group. It was solved, in the a rmative, by von Neumann [23] for compact groups in 1933, and by Gleason [11] and by Montgomery and Zippin [20] for locally compact groups in 1952. A more general version of the Hilbert 5th problem, called the Hilbert-Smith Conjecture, asserts that among all locally compact groups only Lie groups G can act e ectively on ( nite-dimensional) manifolds M (i.e. each g ∈ G\{e} moves at least one point of M) [28]. It follows from the work of Newman [24] and Smith [29] that this conjecture is equivalent to its special case when the acting group G is the group of p-adic integers Ap. In 1946 Bochner and Montgomery [3] proved the Hilbert-Smith Conjecture for groups G acting e ectively on a manifold M by di eomorphisms. A simpler, geometrical proof was obtained by Skopenkov and the authors [25] using the idea of smooth homogeneity: a compact subset K ⊂ M of a smooth manifold M is said to be smoothly ambiently homogeneous, i.e. for each x; y ∈ K there exists a di eomorphism h : (M;K; x)→ (M;K; y). It was shown that this property implies that K is a smooth submanifold of M (therefore G ∼= K is a Lie group). The proof reveals a close relationship between homogeneity and taming theory for compact subsets of Rn, which are pinched by tangent balls (the latter problem was investigated in the past by various authors [6,10,12,16,17]). See also a very interesting paper by Hahn [13]. An interesting approach to the Hilbert-Smith conjecture is via wild Cantor sets in Rn with strong homogeneity properties. Note that the Antoine necklace