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

## TL;DR

This paper extends classical homotopy comparison theorems to o-minimal structures, showing that under certain conditions, the homotopy groups of definable sets match those of their hyperdefinable quotients, with applications to definably compact groups.

## Contribution

It introduces a Vietoris-Smale type theorem for o-minimal homotopy, linking definable sets with hyperdefinable quotients, and provides new proofs of key homotopy comparison results, including Pillay's group conjecture.

## Key findings

- Homotopy groups of definable sets are isomorphic to those of hyperdefinable quotients.
- Dimension equality between definable sets and their quotients.
- New proofs of classical and o-minimal homotopy comparison theorems.

## 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.