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 → G 00 → G → G / G 00 → 1, where G 00 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 / G 00 , 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 / G 00 . We apply this result to show that there is a natural exact sequence1 → G 00 → G ~ →G / G 00~ → 1 , whereG ~is the (o‐minimal) universal cover of G , andG / G 00~is the universal cover of the real Lie group G / G 00 . We also prove that, up to isomorphism, each finite covering H → G / G 00 , with H a connected Lie group, is of the form H / H 00 → G / G 00 for some definable group extension H → G . Finally we prove that the (Lie‐)isomorphism type of G / G 00 determines the definable homotopy type of G . In the semisimple case a stronger result holds: G / G 00 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 / G 00 .