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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.

MSC:

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

References:

[1] DOI: 10.1016/j.apal.2005.01.002 · Zbl 1068.03033 · doi:10.1016/j.apal.2005.01.002
[2] Sheaf theory (1997)
[3] O-minimal spectra, infinitesimal subgroups and cohomology 72 pp 1177– (2007)
[4] Proceedings of the American Mathematical Society 136 pp 1087– (2008)
[5] Sheaves of continuous definable functions 53 pp 1165– (1988)
[6] DOI: 10.1142/S0219061304000346 · Zbl 1069.03029 · doi:10.1142/S0219061304000346
[7] DOI: 10.1016/0022-4049(88)90125-9 · Zbl 0662.03025 · doi:10.1016/0022-4049(88)90125-9
[8] DOI: 10.1090/S0002-9947-00-02593-9 · Zbl 0952.03046 · doi:10.1090/S0002-9947-00-02593-9
[9] DOI: 10.4064/fm193-2-4 · Zbl 1117.03042 · doi:10.4064/fm193-2-4
[10] Journal of the American Mathematical Society 21 pp 563– (2008)
[11] DOI: 10.1142/S0219061304000358 · Zbl 1070.03025 · doi:10.1142/S0219061304000358
[12] DOI: 10.1142/S0219061306000566 · Zbl 1120.03024 · doi:10.1142/S0219061306000566
[13] Forking and independence in o-minimal theories 69 pp 215– (2004)
[14] Journal für die reine und angewandte Mathematik 355 pp 108– (1985)
[15] DOI: 10.1016/0022-4049(83)90058-0 · Zbl 0525.14015 · doi:10.1016/0022-4049(83)90058-0
[16] DOI: 10.1093/qmath/hah010 · Zbl 1065.03020 · doi:10.1093/qmath/hah010
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