[Paper Review] Higher homotopy of groups definable in o-minimal structures

Alessandro Berarducci, Marcello Mamino+1 more|ArXiv.org|Sep 29, 2008

Advanced Topology and Set Theory3 references4 citations

TL;DR

This paper establishes that the o-minimal higher homotopy groups of definably compact groups are isomorphic to those of their associated compact Lie groups, proving that all abelian definably compact groups of the same dimension are definably homotopy equivalent and that their universal covers are contractible. The result extends classical homotopy theory to o-minimal structures using model-theoretic tools and transfers classical topology results to the o-minimal setting.

ABSTRACT

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.

Motivation & Objective

• To extend classical homotopy theory to o-minimal structures by studying higher homotopy groups of definably compact groups.
• To establish that o-minimal higher homotopy groups of definably compact groups are isomorphic to those of their associated compact Lie groups.
• To prove that all abelian definably compact groups of the same dimension are definably homotopy equivalent.
• To show that the universal cover of an abelian definably compact group is contractible in the category of locally definable spaces.

Proposed method

• Use of o-minimal H-space structure and transfer theorems from [1] to relate o-minimal and classical homotopy groups.
• Leverage the fact that o-minimal higher homotopy groups are divisible and prove finite generation to show triviality in the abelian case.
• Apply the o-minimal Whitehead theorem from [1] and [2] to deduce contractibility of universal covers.
• Use the long exact homotopy sequence for fibrations and the structure of centerless and semisimple groups to reduce the non-abelian case to known results.
• Utilize the isomorphism between o-minimal and classical homotopy groups via results in [1], particularly for semialgebraic groups over real algebraic numbers.
• Apply the o-minimal Poincaré-Hurewicz theorem and cohomological isomorphisms from [4] to support the homotopy isomorphism result.

Experimental results

Research questions

• RQ1 Do the o-minimal higher homotopy groups of a definably compact group G coincide with those of its quotient G/G⁰⁰?
• RQ2 Are all abelian definably compact groups of the same dimension definably homotopy equivalent?
• RQ3 Is the universal covering group of an abelian definably compact group contractible in the category of locally definable spaces?
• RQ4 Does the isomorphism between o-minimal and classical homotopy groups extend to higher homotopy groups of semialgebraic groups?
• RQ5 Can the structure of o-minimal H-spaces be used to transfer classical homotopy finiteness results to the o-minimal setting?

Key findings

• The o-minimal higher homotopy groups πₙ(G) of any definably compact group G are isomorphic to the classical higher homotopy groups πₙ(G/G⁰⁰) of the associated compact Lie group.
• For abelian definably compact groups, all higher homotopy groups πₙ(G) vanish for n > 1, implying they are definably homotopy equivalent to a torus of the same dimension.
• The universal covering group of an abelian definably compact group is contractible in the category of locally definable spaces, as all its homotopy groups are trivial.
• The result extends to non-abelian groups via reduction to the semisimple case, using fibrations and the center, with πₙ(G) ≅ πₙ(G/G⁰⁰) holding for all n ≥ 1.
• The isomorphism πₙ(G(M)) ≅ πₙ(G(ℝ)) holds for semialgebraic groups over real algebraic numbers, enabling transfer from o-minimal to classical topology.
• The cohomological isomorphism result from [4] supports the homotopy isomorphism, particularly through the universal coefficient theorem and finite-dimensionality of homology over a field.