A remark on Borsuk’s question on homotopy domination by polyhedra
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 $X$.