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Splitting definably compact groups in o-minimal structures. (English) Zbl 1247.03062

Summary: An argument of A. Borel [Tohoku Math. J., II. Ser. 13, 216–240 (1961; Zbl 0109.26101), Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

MSC:

03C64 Model theory of ordered structures; o-minimality
55S40 Sectioning fiber spaces and bundles in algebraic topology

Citations:

Zbl 0109.26101
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References:

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