2020/06/11 by Daniel Kasprowski, Mark Powell, Kasprowski, Daniel +3
Mathematics · #57K40 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2006.06127
openalex publication_date 2020/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of S2 × S2. 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 \mathbbCP2-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 τ-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 ℤ × ℤ/2.