Topology Seminar, W. W. Comfort (Wesleyan Emeritus): 'Counting Compact Group Topologies'

Wednesday, November 19, 2014
4:15 PM - 5:15 PM (ET)
Event Type
Academic Calendar
Karen Collins
Math CS Topology et al. Seminar

Abstract: Given a group K let 𝔠𝔤𝔱 (K ) be the set of Hausdorff compact group topologies on K. The authors ask: when |K | = κ ≥ ω, what are the possible cardinalities of a pairwise homeomorphic subset (h) ⊆ 𝔠𝔤𝔱 (K ) [resp., pairwise nonhomeomorphic subset 𝕍(n) ⊆ 𝔠𝔤𝔱 (K )]? Revisiting (sometimes improving) therorems of Halmos, Hulanicki, Fuchs, Hawley, Chuan/Liu and Kirku, the authors show inter alia:Always |𝕍 (h )| ≤ 2κ and |𝕍 (h )| ≤ κ.If K is abelian and some 𝖳 ∈ 𝔠𝔤𝔱 (K ) is connected, then |𝕍 (h )| = 2κ does occur. In particular for λ ≥ ω and K = ℝλ or K = 𝕋λ, |𝕍 (h )| = 2(2λ) does occur.[K not necessarily abelian] If some 𝖳 ∈ 𝔠𝔤𝔱 (K ) is connected and the connected component Z0 ( K, 𝖳) of the center of ( K, 𝖳) satisfies π1(Z0 ( K, 𝖳)) ≠ {0}, then |𝔠𝔤𝔱 (K)| ≥ 2𝔠.Corollary to 3: Every nonsemisimple compact connected Lie group admits exactly 2𝔠-many compact group topologies.For K = 𝕋: |𝕍(n)| = 𝔠 occurs in ZFC.For K = ℝ: |(n)| = ω occurs in ZFC; |𝕍(n)| = ω is best possible in ZFC+CH; and |𝕍(n)| > ω is consistent with ZFC.*Joint work with Dieter Remus

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