Barratt–Priddy–Quillen theorem

Remarks

Barratt and Priddy, and independently Quillen, proved the theorem in the early 1970s. One proof uses the group completion theorem: the homology of the group completion of $M = \coprod_n B\Sigma_n$ is $H_*(M)[\pi_0(M)^{-1}] = \Z[\Z] \otimes H_*(B\Sigma_\infty)$. The other input is that $\coprod_n B\Sigma_n$ is the free $E_\infty$ space on a point, and that the group completion of the free $E_\infty$ space on $S^0$ is $QS^0$.

At work: taking homotopy groups gives $\pi_k(B\Sigma_\infty^+) \cong \pi_k^s$ for $k \ge 1$, the stable homotopy groups of spheres. For $k = 1$ this says $\pi_1^s \cong \Z/2$, which matches the abelianization $\Sigma_\infty / A_\infty \cong \Z/2$, where $A_\infty$ is the infinite alternating group and the quotient map is the sign of a permutation.

Going deeper

For a student

Since $\pi_1(B\Sigma_\infty^+)$ is the abelianization of $\Sigma_\infty$, the theorem recovers $\pi_1^s \cong \Z/2$, and the higher groups give a description of all stable stems in terms of permutations. Together with Nakaoka's homological stability for symmetric groups, it computes $H_*(Q_0 S^0)$ from the homology of the groups $\Sigma_n$. Quillen's analogue for general linear groups over finite fields led to his computation of the algebraic K-theory of finite fields.