Detecting homotopy equivalences in base-point-free homotopy

Abstract

Let ƒ:X→Y be a map of connected CW complexes, such that ƒ#:[K, X]→[K, Y] is a bijection for every finite complex K ([K, X] being the set of free homotopy classes). Examples are known where such ƒ is not a homotopy equivalence, but no example is known where Y is finite-dimensional. We prove that if Y has finite dimension d and if HdỸ ≠ 0 then ƒ is a homotopy equivalence. More generally, we give necessary and sufficient conditions in terms of the fundamental group of Y for ƒ to be a homotopy equivalence, and we show that these conditions are almost met whenever Y is finite-dimensional. An interesting sequence {Gm}m⩾2 of finitely presented ‘Artin groups’ appears naturally in the discussion.