Title: 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 $\pi_{n}(X)^{\rm
def}\cong\pi_{n}(X/E)$ 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.