Homotopy epimorphisms in homotopy pullbacks (Q1336619)

The authors prove that homotopy epimorphisms are preserved under homotopy pullback. (A map \(f: X\to Y\) of pointed path-connected CW-spaces is a homotopy epimorphism, if given \(u,v: Y\to Z\), \(u\circ f\simeq v\circ f\) implies \(u\simeq v\).) The proof makes use of \textit{M. Mather}'s first cube theorem [Can. J. Math. 28, 225-263 (1976; Zbl 0351.55005)]. As an application a generalization of a result of \textit{J. Dydak} on homotopy epimorphisms between universal covering spaces [Proc. Am. Math. Soc. 116, No. 4, 1171-1173 (1992; Zbl 0768.55007)]\ is given.