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 up to stabilisation by connected sums with copies of S² × S². This stable classification is detected by a spin bordism group over the classifying space Bπ 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 π₂. In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups.