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/G00 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 → e G → G̃/G00 → 1 where e G is the (o-minimal) universal cover of G, and G̃/G00 is the universal cover of the real Lie group G/G00. We also prove that, up to isomorphism, each finite extension H of G/G00 in the category of Lie groups is of the form H/H00 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.