A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets

2017/06/07 by Achille, Alessandro, Berarducci, Alessandro
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1706.02094

Abstract

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)^{\rm def}≅π_n(X/E) and dim(X)=dim_{\mathbb 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_{\mathbb R}(G/G⁰⁰)" largely independent of the group structure of G. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.

Related

• On o-minimal homotopy groups
• Locally definable homotopy
• On the o-minimal LS-category
• Definable groups as homomorphic images of semilinear and field-definable groups
• Higher homotopy of groups definable in o-minimal structures