On isometries of symmetric products of metric spaces
On isometries of symmetric products of metric spaces
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关于度量空间对称积的等距
DOI：
10.1016/j.topol.2018.08.006
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发表时间：
2018
影响因子：
0.6
通讯作者：
Naotsugu Chinen
中科院分区：
文献类型：
--
作者：
Norihiko Minami;Naotsugu Chinen
设度量空间（X，d）的n阶对称积Fn（X），n≥ 1表示X的至多n个元素具有Hausdorff度量dH的非空有限子集空间.用Iso（X）表示从X到自身的所有等距的群，具有点态收敛的拓扑。本文证明了在一定的假设下，Iso（Fn（X））拓扑同构于半直积群Iso（Fn（X），F1（X））<$Iso（X）.将这些结果应用于特殊空间<$pq，（p，q）∈[1，∞]× N≥ 2 <$$>，证明了：（1）若p∈{1，∞}，则Iso（F2（<$p2））拓扑同构于Z2 × Iso（<$p2）. (2)若3≤ q<∞，则Iso（F2（<$∞ q））拓扑同构于<$i= 1 q− 1（Z2）i <$Iso（<$∞ q）。(3)在除（n，p，q）∈ N≥ 2×{1，∞}×{∞}以外的其它情形下，标准同态χ n：Iso（n，p，q）→ Iso（Fn（n，p，q））是拓扑同构.
By F n (X), n≥ 1, we denote the n-th symmetric product of a metric space (X, d) as the space of the nonempty finite subsets of X with at most n elements endowed with the Hausdorff metric d H. By Iso (X) we denote the group of all isometries from X onto itself with the topology of pointwise convergence. In this paper, we show that, under the certain hypothesis, Iso (F n (X)) is topologically isomorphic to the semidirect product group Iso (F n (X), F 1 (X))⋊ Iso (X). We apply those results to ℓ p q,(p, q)∈[1,∞]× N≥ 2⁎, as particular spaces and prove the following statements:(1) If p∈{1,∞}, then Iso (F 2 (ℓ p 2)) is topologically isomorphic to Z 2× Iso (ℓ p 2).(2) If 3≤ q<∞, then Iso (F 2 (ℓ∞ q)) is topologically isomorphic to∏ i= 1 q− 1 (Z 2) i⋊ Iso (ℓ∞ q).(3) In other cases except (n, p, q)∈ N≥ 2×{1,∞}×{∞}, the canonical homomorphism χ n: Iso (ℓ p q)→ Iso (F n (ℓ p q)) is a topological isomorphism.