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

摘要

We provide an example of a Q-homology equivalence f : X -> Y between CW-complexes X and Y with finitely generated integral homology groups such that f is not a Z(p)-homology equivalence for some prime p which divides no torsion coefficients of H-i(X) and H-i(Y) for all i is an element of 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.