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, π₁(P) may be any given non-abelian poly-ℤ-group and dim P may be any given integer n ≥ 3.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-1,
author = {Danuta Ko\l odziejczyk},
title = {Homotopy dominations within polyhedra},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {189-202},
zbl = {1060.55003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-1}
}
Danuta Kołodziejczyk. Homotopy dominations within polyhedra. Fundamenta Mathematicae, Tome 177 (2003) pp. 189-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-1/