Counting compact group topologies

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• 作者：W.W. Comfort^a ; ^b ; ^{wcomfort@wesleyan.edu" class="auth_mail" title="E-mail the corresponding author} ; Dieter Remus^c ; ^{remus@math.uni-paderborn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 22H11 ; secondary ; 54A25
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：213
• 期：Complete
• 页码：92-109
• 全文大小：550 K

文摘

Given a group K  , the symbol cgt(K)$\mathfrak{cgt}(K)$ denotes the set of Hausdorff compact group topologies on K  . The authors ask: when |K|=κ≥ω$|K|=\kappa \ge \omega$, what are the possible cardinalities of a pairwise homeomorphic subset [resp., a pairwise nonhomeomorphic subset] of cgt(K)$\mathfrak{cgt}(K)$? Revisiting (sometimes, improving) theorems of Halmos, Hulanicki, Fuchs, Hawley, Chuan/Liu, Kirku, and Shtern, the authors show inter alia:

1. |cgt(K)|≤2^κ$|\mathfrak{cgt}(K)|\le 2^{\kappa }$, and every pairwise nonhomeomorphic V⊆cgt(K)$\mathtt{V}\subseteq \mathfrak{cgt}(K)$ satisfies 7490751704ee3d974a2d" title="Click to view the MathML source">|V|≤κ$|\mathtt{V}|\le \kappa$.

2. If K   is infinite, abelian and divisible with cgt(K)≠∅$\mathfrak{cgt}(K)\ne \varnothing$, there is a pairwise homeomorphic V⊆cgt(K)$\mathtt{V}\subseteq \mathfrak{cgt}(K)$ such that |V|=|cgt(K)|=2^κ$|\mathtt{V}|=|\mathfrak{cgt}(K)|=2^{\kappa }$. In particular for λ≥ω$\lambda \ge \omega$ and K=R^λ$K={\mathbb{R}}^{\lambda }$ or K=T^λ$K={\mathbb{T}}^{\lambda }$, there is a pairwise homeomorphic V⊆cgt(K)$\mathtt{V}\subseteq \mathfrak{cgt}(K)$ such that |V|=2^|K|=2^(2λ)$|\mathtt{V}|=2^{|K|}=2^{(2^{\lambda })}$.

3. [K   not necessarily abelian] If some T∈cgt(K)$\mathcal{T}\in \mathfrak{cgt}(K)$ is connected and the connected component Z₀(K,T)$Z_0(K,\mathcal{T})$ of the center of (K,T)$(K,\mathcal{T})$ satisfies π₁(Z₀(K,T))≠{0}${\pi }_1(Z_0(K,\mathcal{T}))\ne \{0\}$, then |cgt(K)|=2^|K|$|\mathfrak{cgt}(K)|=2^{|K|}$.

4. Corollary to 3: Every nonsemisimple compact connected Lie group (K,T)$(K,\mathcal{T})$ satisfies |cgt(K)|=2^c$|\mathfrak{cgt}(K)|=2^{\mathfrak{c}}$.

5. [Hulanicki] In ZFC: there is a pairwise nonhomeomorphic V⊆cgt(T)$\mathtt{V}\subseteq \mathfrak{cgt}(\mathbb{T})$ such that |V|=c$|\mathtt{V}|=\mathfrak{c}$.

6. Concerning pairwise nonhomeomorphic V⊆cgt(R)$\mathtt{V}\subseteq \mathfrak{cgt}(\mathbb{R})$: |V|=ω$|\mathtt{V}|=\omega$ occurs in ZFC; |V|=ω$|\mathtt{V}|=\omega$ is best possible in ZFC + CH; and |V|>ω$|\mathtt{V}|>\omega$ is consistent with ZFC.

7. A compact abelian group (K,T)$(K,\mathcal{T})$ satisfies cgt(K)={T}$\mathfrak{cgt}(K)=\{\mathcal{T}\}$ if and only if each automorphism of K   is a 120" class="mathmlsrc">120.gif&_user=111111111&_pii=S016686411630181X&_rdoc=1&_issn=01668641&md5=c4d36dcf758df44c6a33dcbb38b8787e" title="Click to view the MathML source">T-homeomorphism.