Higher homotopy of groups definable in o-minimal structures (Q616515)

It is known that every definably compact group \(G\) has a canonical type-definable normal subgroup \(G^{00}\) such that the factor \(G/G^{00}\) with a ``logic topology'' is a compact Lie group. Moreover, \(G^{00}\) is divisible torsion-free. In a joint paper with M. Edmundo, the third author studied the o-minimal fundamental group \(\pi_1(G)\) for an abelian \(G\) and showed that it is isomorphic to \(\pi(G/G^{00})\) -- the fundamental group of the Lie group \(G/G^{00}\). The first author extended this result to the non-abelian case. In this paper, the o-minimal \(n\)-th homotopy group \(\pi_n(G)\) of \(G\) and the \(n\)-th homotopy group \(\pi(G/G^{00})\) of \(G/G^{00}\) are compared for all \(n > 1\) and all definably compact groups \(G\). It is proved that, just as in the case \(n = 1\), they are isomorphic for every \(n\). In the abelian case this amounts to checking that \(\pi_n(G) = 0\), and indeed to showing that \(\pi_n(G)\) is finitely generated (since \(\pi_n (G)\) is divisible). This is done by using the o-minimal \(H\)-space of the triangulated copy of \(G\) and results by E. Baro and the third author linking the classical theory of \(H\)-spaces and the o-minimal setting. As a consequence, it is shown that all the abelian definably compact definably connected groups of the same dimension are definably homotopy equivalent. The analysis for non-abelian groups is reduced to the study of the abelian and the semisimple centreless cases, making use of the long exact homotopy group of a fibration and again of results by Baro and the third author.