Real Algebraic and Analytic Geometry |

Equivariant homotopy of definable groups.

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Submission: 2009, May 11.

*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/G^{00} 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/G^{00}.
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/G^{00} 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/G^{00} to G'/G'^{00} induces a
definable isomorphism from G to G'.*

Mathematics Subject Classification (2000): 03C64, 03H05, 22E15.

Keywords and Phrases: Homotopy, Definable groups, o-minimality.

**Full text**: http://arxiv.org/abs/0905.1069

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