Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup. (English) Zbl 1181.03042

Let \(M\) be a saturated o-minimal expansion of an ordered field. Recent results, related to conjectures of Pillay, prove that each definably compact group \(G\) in \(M\) is an extension of a compact Lie group by a torsion-free normal divisible subgroup \(G^{00}\), called the infinitesimal subgroup of \(G\). Here it is shown that this subgroup is cohomogically acyclic. The proof reduces the general analysis to the abelian and the definably simple cases. The former is handled by the abelian case of the compact domination conjecture.
As a consequence, a canonical isomorphism is obtained between the o-minimal cohomology of a definably compact group and the cohomology of the associated Lie group.


03C64 Model theory of ordered structures; o-minimality
03H05 Nonstandard models in mathematics
22E15 General properties and structure of real Lie groups
Full Text: DOI arXiv


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