Abstract

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.

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