We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated in a functorial way an intrinsic "infinitesimal subgroup" G(00) and a real Lie group G/G(00). We prove that that the isomorphism type of G/G(00) in the Lie category determines the definable homotopy type of G. In the semisimple case a stronger result holds, namely the definable isomorphism type of G is determined by the associated Lie group. The proof depends on the study of the homotopy properties of the projection of G onto the associated Lie group. It is shown in particular that the preimage of a simply connected open set is simply connected in the definable category. This will also allow us to show that there is a correspondence between finite group extensions in the Lie category and in the definable category.
On the homotopy type of definable groups in an o-minimal structure
BERARDUCCI, ALESSANDRO;
2011-01-01
Abstract
We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated in a functorial way an intrinsic "infinitesimal subgroup" G(00) and a real Lie group G/G(00). We prove that that the isomorphism type of G/G(00) in the Lie category determines the definable homotopy type of G. In the semisimple case a stronger result holds, namely the definable isomorphism type of G is determined by the associated Lie group. The proof depends on the study of the homotopy properties of the projection of G onto the associated Lie group. It is shown in particular that the preimage of a simply connected open set is simply connected in the definable category. This will also allow us to show that there is a correspondence between finite group extensions in the Lie category and in the definable category.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
