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 a natural exact sequence 1 â G00 â G â G/G00 â 1, where G00 is the 'infinitesimal subgroup' of G and G/G 00 is a compact real Lie group. We show that given a connected open subset U of G/G00, there is a canonical isomorphism between the fundamental group of U and the o-minimal fundamental group of its preimage under the projection p: Gâ G/G00. We apply this result to show that there is a natural exact sequence 1âG00âGÌ â G/G00â1, where GÌ is the (o-minimal) universal cover of G, and G/G00 is the universal cover of the real Lie group G/G 00. We also prove that, up to isomorphism, each finite covering H â G/G00, with H a connected Lie group, is of the form H/H 00â G/G00 for some definable group extension HâG. Finally we prove that the (Lie-)isomorphism type of G/G00 determines the definable homotopy type of G. In the semisimple case a stronger result holds: G/G00 determines G up to definable isomorphism. Our results depend on the study of the o-minimal fundamental groupoid of G and the homotopy properties of the projection Gâ G/G00. © 2011 London Mathematical Society.
On the homotopy type of definable groups in an o-minimal structure
Berarducci, Alessandro;Mamino, Marcello
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 a natural exact sequence 1 â G00 â G â G/G00 â 1, where G00 is the 'infinitesimal subgroup' of G and G/G 00 is a compact real Lie group. We show that given a connected open subset U of G/G00, there is a canonical isomorphism between the fundamental group of U and the o-minimal fundamental group of its preimage under the projection p: Gâ G/G00. We apply this result to show that there is a natural exact sequence 1âG00âGÌ â G/G00â1, where GÌ is the (o-minimal) universal cover of G, and G/G00 is the universal cover of the real Lie group G/G 00. We also prove that, up to isomorphism, each finite covering H â G/G00, with H a connected Lie group, is of the form H/H 00â G/G00 for some definable group extension HâG. Finally we prove that the (Lie-)isomorphism type of G/G00 determines the definable homotopy type of G. In the semisimple case a stronger result holds: G/G00 determines G up to definable isomorphism. Our results depend on the study of the o-minimal fundamental groupoid of G and the homotopy properties of the projection Gâ G/G00. © 2011 London Mathematical Society.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.