拓扑群的一般性质
Generic properties of topological groups
AI总结：
本文利用Baire范畴方法研究拓扑群的一般性质，证明可数无限离散阿贝尔群和紧可度量化阿贝尔群均存在剩余同构类，并刻画了泛型紧可度量化群的连通性与挠性质。
摘要：
我们研究拓扑群在Baire范畴意义下的一般性质。首先，我们考察可数无限（离散）群。我们推广了B. H. Neumann、H. Simmons和A. Macintyre关于代数闭群和字问题的经典结果。I. Goldbring、S. E. Kunnawalkam和Y. Lodha证明了在可数无限（离散）群中，每个同构类都是贫乏的。与之相反，我们证明了在可数无限（离散）阿贝尔群中存在一个剩余同构类。然后我们转向紧可度量化阿贝尔群。我们利用Pontryagin对偶性证明了在紧可度量化阿贝尔群中存在一个剩余同构类。我们讨论了它与可数无限（离散）情形的联系。最后，我们研究紧可度量化群。我们证明了泛型紧可度量化群既不是连通的，也不是完全不连通的；它既不是无挠的，也不是挠群。
英文摘要：
We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. I. Goldbring, S. E. Kunnawalkam and Y. Lodha proved that every isomorphism class is meager among countably infinite (discrete) groups. In contrast, we show that there is a comeager isomorphism class among countably infinite (discrete) abelian groups. Then we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite (discrete) case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.
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