Algebraic criteria for stable diffeomorphism of spin 4-manifolds

Kasprowski, D., Powell, M. and Teichner, P. (2023) Algebraic criteria for stable diffeomorphism of spin 4-manifolds. Memoirs of the American Mathematical Society, (Accepted for Publication)

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 $\tau$-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 $\mathbb{Z} \times \mathbb{Z}/2$.

• Item Type: Articles
• Additional Information: The authors thank the Max Planck Institute for Mathematics and the Hausdorff Institute for Mathematics in Bonn for financial support and their excellent research environments. The second author was partially supported by EPSRC New Investigator grant EP/T028335/2 and EPSRC New Horizons grant EP/V04821X/2.
• Status: Accepted for Publication
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Powell, Professor Mark
• Authors: Kasprowski, D., Powell, M., and Teichner, P.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Memoirs of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0065-9266
• ISSN (Online): 1947-6221

Related URLs

• Publisher