Plus construction

Let $X$ be a space and $H⊲{\pi }_1(X)$ a normal subgroup that is perfect (i.e. $[H,H]=H$). The plus construction is the essentially unique space $X^{+}$ with fundamental group ${\pi }_1(X)/H$ and equipped with a map $X\to X^{+}$ which induces an isomorphism on homology for all coefficients. It is useful in situations where one is given a map $A\to B$ which is an isomorphism on homology groups but acts wildly on homotopy groups; in some situations, applying the plus construction can replace this homology equivalence with a homotopy equivalence—this is how the plus construction plays a role in group completion.