Epimorphisms and monomorphisms in homotopy

Abstract:

The main result of this note is the following: Theorem A. If $f:X \to Y$ is an epimorphism of $\mathcal {H}\mathcal {C}{\mathcal {W}^*}$, the homotopy category of pointed path-connected CW-spaces, and ${\pi _1}(f):{\pi _1}(X) \to {\pi _1}(Y)$ is a monomorphism, then $\tilde f:\tilde X \to \tilde Y$ is an epimorphism of $\mathcal {H}\mathcal {C}{\mathcal {W}^*}$. As a straightforward consequence the following results of Dyer-Roitberg (Topology Appl. (to appear)) is derived: Theorem B. A map $f:X \to Y$ is an equivalence in $\mathcal {H}\mathcal {C}{\mathcal {W}^*}$, the homotopy category of pointed path-connected CW-spaces, iff it is both an epimorphism and a monomorphism in $\mathcal {H}\mathcal {C}{\mathcal {W}^*}$.