Plus construction

Let $ X$ be a space and $ H \vartriangleleft \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 \rightarrow X^{+}$ which induces an isomorphism on homology for all coefficients. It is useful in situations where one is given a map $ A \rightarrow 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.

Jeffrey Herschel Giansiracusa 2005-06-27