Quillen plus construction (Ex)

Latest revision as of 05:44, 8 January 2019

Let $<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 * $H_*f$ is an isomorphism. * $\pi_1(f) : \pi_1X \to \pi_1X_+ = (\pi_1 X)/H$. In fact, this is unique up to homotopy. </wikitex> [[Category:Exercises]] [[Category:Exercises without solution]]X$ be a connected CW-complex. Let $H = [H,H] \triangleleft \pi_1X$. Show there exists a map $f : X \to X_+$ so that

• $H_*f$ is an isomorphism.
• $\pi_1(f) : \pi_1X \to \pi_1X_+ = (\pi_1 X)/H$.

In fact, this is unique up to homotopy.