Dear community,

assume that $f\colon X\to Y$ is a homology equivalence, i.e. induces isomorphisms on all integer homology groups. Is there a map $g\colon Y\to X$ which is also a homology equivalence?

Actually, I am only interested in the case if $f\colon X\to X^+$ is obtained by Quillen´s plus construction.

Thanks for any comments in advance.