Quillen plus construction (Ex)

Let $X$$<wikitex>; Let $X$ be a connected CW-complex. Let $H = [H,H] \triangleleft \pi_1X$. Show there exists a map $f : X \to X_+$ so that \begin{itemize} \item $H_*f$ is an isomorphism. \item $\pi_1(f) : \pi_1X \to \pi_1X_+ = \pi_1 X/H$. \end{itemize} </wikitex> [[Category:Exercises]] [[Category:Exercises without solution]]X$ be a connected CW-complex. Let $H = [H,H] \triangleleft \pi_1X$$H = [H,H] \triangleleft \pi_1X$. Show there exists a map $f : X \to X_+$$f : X \to X_+$ so that \begin{itemize} \item $H_*f$$H_*f$ is an isomorphism. \item $\pi_1(f) : \pi_1X \to \pi_1X_+ = \pi_1 X/H$$\pi_1(f) : \pi_1X \to \pi_1X_+ = \pi_1 X/H$. \end{itemize}