# Plus construction

Let be a space and a normal subgroup that is perfect (i.e. ). The plus construction is the essentially unique space with fundamental group and equipped with a map which induces an isomorphism on homology for all coefficients. It is useful in situations where one is given a map 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