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Algebraic criteria for stable diffeomorphism of spin 4-manifolds

Daniel Kasprowski, Mark Powell, Peter Teichner

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

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 . 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 -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 -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 -manifolds with fundamental group .

Paper Structure

This paper contains 43 sections, 98 theorems, 328 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The stable classification for fundamental group ZxZ/2
  3. Review of stable classification
  4. Translating the bordism invariants into algebra
  5. The algebraic primary obstruction
  6. The algebraic secondary obstruction
  7. The algebraic tertiary obstruction
  8. Context for our work
  9. When stable homeomorphism implies homeomorphism
  10. Background
  11. Stable diffeomorphism, bordism groups, and spectral sequences
  12. Bordism groups
  13. Spectral sequences
  14. Topological 4-manifolds and stable homeomorphism
  15. Whitehead's certain exact sequence and the Gamma functor
  16. ...and 28 more sections

Key Result

Theorem 1.1

Fix a group $\pi$ with finite 3-dimensional classifying space, or an abelian group, a right angled Artin group, or a finite group $\pi$ with abelian or quaternion $2$-Sylow subgroup. One can decide whether two closed, connected, spin, smooth 4-manifolds $N_1$ and $N_2$ are stably diffeomorphic over

Figures (1)

  • Figure 1: The proof of tertiary inheritance.

Theorems & Definitions (197)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.5
  • Theorem 1.6: Primary obstruction theorem KPT18
  • Definition 1.7
  • Lemma 1.8
  • Remark 1.9
  • Remark 1.10
  • Theorem 1.12
  • ...and 187 more