The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds

We consider the role of the Kervaire--Milnor invariant in the classification of closed, connected, spin 4-manifolds, typically denoted by $M$, up to stabilisation by connected sums with copies of $S^2 \times S^2$. This stable classification is detected by a spin bordism group over the classifying space $B\pi$ of the fundamental group. Part of the computation of this bordism group via an Atiyah--Hirzebruch spectral sequence is determined by a collection of codimension two Arf invariants. We show that these Arf invariants can be computed by the Kervaire--Milnor invariant evaluated on certain elements of $\pi_2(M)$. In particular this yields a new stable classification of spin $4$-manifolds with 2-dimensional fundamental groups, namely those for which $B\pi$ admits a finite 2-dimensional CW-complex model.