Homotopy dominations within polyhedra
Volume 178 / 2003
Fundamenta Mathematicae 178 (2003), 189-202
MSC: 55P55, 55P15.
DOI: 10.4064/fm178-3-1
Abstract
We show the existence of a finite polyhedron $P$ dominating infinitely many different homotopy types of finite polyhedra and such that there is a bound on the lengths of all strictly descending sequences of homotopy types dominated by $P$. This answers a question of K. Borsuk (1979) dealing with shape-theoretic notions of “capacity” and “depth” of compact metric spaces. Moreover, $\pi _1(P)$ may be any given non-abelian poly-${{\mathbb Z}}$-group and $\mathop {\rm dim}\nolimits P$ may be any given integer $n \geq 3$.
