Plus construction

Remarks

A plus construction exists for every such pair $(X, P)$, and it is unique up to homotopy equivalence under $X$. One builds it by attaching $2$-cells along loops that represent generators of $P$, and then $3$-cells that cancel the new classes in degree $2$. The perfectness of $P$ is what makes the second step possible. Every map $X \to Y$ that kills $P$ on $\pi_1$ factors through $X^+$, uniquely up to homotopy.

Example: let $R$ be a ring and $X = BGL(R)$, the classifying space of the discrete group $GL(R)$. By Whitehead's lemma, the elementary subgroup $E(R)$ is perfect and equals $[GL(R), GL(R)]$. The plus construction for $P = E(R)$ gives $BGL(R)^+$, with $\pi_1(BGL(R)^+) = GL(R)/E(R) = K_1(R)$, and Quillen defines $K_n(R) = \pi_n(BGL(R)^+)$ for $n \ge 1$.

Non-example: take $X = S^1$ and $P = \pi_1(S^1) \cong \Z$, which is not perfect. A map $f \colon S^1 \to Y$ with clause 1 makes $\pi_1(Y)$ trivial, so $H_1(Y; \Z) = 0$ by the Hurewicz theorem. But $H_1(S^1; \Z) = \Z$, so clause 2 fails for $M = \Z$.