Algebraic criteria for stable diffeomorphism of spin 4-manifolds

Abstract

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2\times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects ${\mathbb{CP}}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors. In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $\ddot{I}„$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z}\times \mathbb{Z}/2$.

102 pages. Version 2: Some results on the Kervaire-Milnor invariant have been extracted to create arXiv:2105.12153. A new Chapter 7 gives an application of our theory. Version 3: Changes following a referee report. Accepted for publication in Memoirs of the American Mathematical Society