Homotopy dominations within polyhedra

Danuta Kołodziejczyk¹

¹ Faculty of Mathematics and Informational Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 189-202

Résumé

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 $n \geq 3$.

Détail

DOI : 10.4064/fm178-3-1

Keywords: existence finite polyhedron dominating infinitely many different homotopy types finite polyhedra there bound lengths strictly descending sequences homotopy types dominated answers question borsuk dealing shape theoretic notions capacity depth compact metric spaces moreover may given non abelian poly mathbb group mathop dim nolimits may given integer geq