65 research outputs found
Higher homotopy of groups definable in o-minimal structures
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.Comment: 13 pages, to be published in the Israel Journal of Mathematic
A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets
Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy
groups of two topological spaces and whenever a map with
strong connectivity conditions on the fibers is given. We apply similar
techniques in o-minimal expansions of fields to compare the o-minimal homotopy
of a definable set with the homotopy of some of its bounded hyperdefinable
quotients . Under suitable assumption, we show that and . As a special case,
given a definably compact group, we obtain a new proof of Pillay's group
conjecture ")" largely independent of the
group structure of . We also obtain different proofs of various comparison
results between classical and o-minimal homotopy.Comment: 24 page
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