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

Abstract

Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ 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 ${\pi }_n(X{)}^{\mathrm{def}}\cong {\pi }_n(X/E)$ 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 "$\dim (G)={\dim }_{\mathbb{R}}(G/G^{00}$)" largely independent of the group structure of $G$. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.

24 pages