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A remark on Borsuk’s question on homotopy domination by polyhedra

Volume 249 / 2020

Cheol Woo Baek, Jang Hyun Jo Fundamenta Mathematicae 249 (2020), 105-110 MSC: Primary 20J05; Secondary 55N99. DOI: 10.4064/fm475-10-2019 Published online: 11 February 2020

Abstract

We provide an example of a $\mathbb Q $-homology equivalence $f : X \to Y$ between $CW$-complexes $X$ and $Y$ with finitely generated integral homology groups such that $f$ is not a $\mathbb Z _p$-homology equivalence for some prime $p$ which divides no torsion coefficients of $H_i(X)$ and $H_i(Y)$ for all $i \in \mathbb N $. This shows that some affirmative answers to a question of Borsuk’s are not justified. We also study the question when a topological space dominates countably infinitely many different homotopy types. As a result, we show that if $X$ is a quasi-finite nilpotent space, then there are countably many different homotopy types of $CW$-complexes dominated by $X$.

Authors

  • Cheol Woo BaekDepartment of Mathematics
    Sogang University
    Seoul, 121-742 Korea
    e-mail
  • Jang Hyun JoDepartment of Mathematics
    Sogang University
    Seoul, 121-742 Korea
    e-mail

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