On the homotopy type of definable groups in an o-minimal structure

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 → G00 → G → G/G00 → 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: G→ G/G00. We apply this result to show that there is a natural exact sequence 1→G00→G̃ → G/G00→1, where G̃ 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 → G/G00, with H a connected Lie group, is of the form H/H 00→ G/G00 for some definable group extension H→G. 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 G→ G/G00. © 2011 London Mathematical Society.