Submission date: 11 May 2009.

Abstract:

We consider groups definable in an o‐minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G⁰⁰ which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G⁰⁰. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the “compact domination conjecture” proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G⁰⁰ which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to prove the following result. Let G and G' be two definably compact definably connected groups with isomorphic associated Lie groups. Then G and G' are definably homotopy equivalent. Moreover given a finite subgroup Γ of G, there is a definable homotopy equivalence f:G→G' that restricted to Γ is an isomorphism onto its image and such that f(cx)=f(c)f(x) for all c∈Γ and x∈G. In the semisimple case a stronger result holds: any Lie isomorphism from G/G⁰⁰ to G'/G'⁰⁰ induces a definable isomorphism from G to G'.

Mathematics Subject Classification: 03C64, 03H05, 22E15

Keywords and phrases: Homotopy, Definable groups, o-minimality

Full text arXiv: pdf, ps.