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Preprint Number 1231

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1231. Alessandro Achille and Alessandro Berarducci
A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets

Submission date: 7 June 2017


Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map f:X → Y 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 X with the homotopy of some of its bounded hyperdefinable quotients X/E. Under suitable assumption, we show that π_{n}(X)^{def} ≡ π_{n}(X/E) and dim(X)=dim_{R}(X/E). As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture “dim(G)=dim_{R}(G/G^{00})” largely independent of the group structure of G. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.

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Full text arXiv 1706.02094: pdf, ps.

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