EPIMORPHISMS AND MONOMORPHISMS IN HOMOTOPY

The main result of this note is the following: Theorem A. If f: X --> Y is an epimorphism H L W*, the homotopy category of pointed path-connected CW-spaces, and pi1(f):pi1(X) --> pi1(Y) is a monomorphism, then f: X --> Y is an epimorphism of H L W*. As a straightforward consequence the following results of Dyer-Roitberg (Topology Appl. (to appear)) is derived: Theorem B. A map f: X --> Y is an equivalence in H L W*, the homotopy category of pointed path-connected CW-spaces, iff it is both an epimorphism and a monomorphism in H L W*.