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

Abstract

Results of Smale (Proc Am Math Soc 8(3): 604–604, 1957) and Dugundji (Fundam Math 66:223–235, 1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map $f:X\rightarrow Y$ with strong connectivity conditions on the fibers is given. We can apply similar techniques to compare the homotopy of spaces living in different categories, for instance an abelian variety over an algebraically closed field, and a real torus. More generally, working in o-minimal expansions of fields, we 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 $\pi _{n}^{{\text {def}}}(X)\cong \pi _{n}(X/E)$ 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.