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Stable equivalence relations on 4-manifolds

Daniel Kasprowski, John Nicholson, Simona Veselá

TL;DR

The article develops a comprehensive framework for stable equivalence relations among closed oriented smooth 4-manifolds using Kreck's modified surgery, recasting these relations as ξ-bordism classes modulo subgroups of Ω4(ξ). It proves a sharp correspondence between six geometric stable relations and algebraic data in Ω4(ξ), via κ2^h, κ2^s, and the pri/sec/ter invariants arising from the James spectral sequence. A key application shows that closed oriented homotopy equivalent 4-manifolds with abelian fundamental groups are stably diffeomorphic, and it provides algebraic obstructions to the existence of smooth 4-manifolds that are homotopy equivalent but not simple homotopy equivalent up to stabilisation. The paper also determines stable rigidity for several finite groups (abelian, quaternion, dihedral, semi-dihedral, modular maximal-cyclic) and analyzes the interplay between stable and unstable equivalence relations, including cancellation phenomena and their impact on 4-manifold classification across topological and smooth categories.

Abstract

Kreck's modified surgery gives an approach to classifying smooth $2n$-manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of $S^n \times S^n$. In dimension 4, we use a combination of modified and classical surgery to study various stable equivalence relations which we compare to stable diffeomorphism. Most importantly, we consider homotopy equivalence up to stabilisation with copies of $S^2 \times S^2$. As an application, we show that closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic. We give analogues of the cancellation theorems of Hambleton--Kreck for stable homeomorphism for homotopy up to stabilisations. Finally, we give a complete algebraic obstruction to the existence of closed smooth 4-manifolds which are homotopy equivalent but not simple homotopy equivalent up to connected sum with $S^2 \times S^2$.

Stable equivalence relations on 4-manifolds

TL;DR

The article develops a comprehensive framework for stable equivalence relations among closed oriented smooth 4-manifolds using Kreck's modified surgery, recasting these relations as ξ-bordism classes modulo subgroups of Ω4(ξ). It proves a sharp correspondence between six geometric stable relations and algebraic data in Ω4(ξ), via κ2^h, κ2^s, and the pri/sec/ter invariants arising from the James spectral sequence. A key application shows that closed oriented homotopy equivalent 4-manifolds with abelian fundamental groups are stably diffeomorphic, and it provides algebraic obstructions to the existence of smooth 4-manifolds that are homotopy equivalent but not simple homotopy equivalent up to stabilisation. The paper also determines stable rigidity for several finite groups (abelian, quaternion, dihedral, semi-dihedral, modular maximal-cyclic) and analyzes the interplay between stable and unstable equivalence relations, including cancellation phenomena and their impact on 4-manifold classification across topological and smooth categories.

Abstract

Kreck's modified surgery gives an approach to classifying smooth -manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of . In dimension 4, we use a combination of modified and classical surgery to study various stable equivalence relations which we compare to stable diffeomorphism. Most importantly, we consider homotopy equivalence up to stabilisation with copies of . As an application, we show that closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic. We give analogues of the cancellation theorems of Hambleton--Kreck for stable homeomorphism for homotopy up to stabilisations. Finally, we give a complete algebraic obstruction to the existence of closed smooth 4-manifolds which are homotopy equivalent but not simple homotopy equivalent up to connected sum with .
Paper Structure (21 sections, 46 theorems, 55 equations, 2 figures, 3 tables)

This paper contains 21 sections, 46 theorems, 55 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Comparison between stable equivalence relations
  3. Comparison between stable and unstable equivalence relations
  4. Surgery theory for 4-manifolds
  5. Kreck's modified surgery
  6. Methods from classical surgery
  7. The transfer in $L$-theory
  8. Proof of \ref{['thmx:main-table']}
  9. Stable rigidity
  10. General results for (strong) stable rigidity
  11. Stable rigidity for subgroups of odd index
  12. Abelian groups
  13. Quaternion groups
  14. Dihedral and semi-dihedral groups
  15. Dihedral groups
  16. ...and 6 more sections

Key Result

Theorem 1

Let $M$, $M'$ be closed, oriented, almost spin, smooth $4$-manifolds with normal $1$-type $\xi=\xi(\pi,w)$. For each subgroup $A \le \Omega_4(\xi)$ listed below, the manifolds $M$ and $M'$ are $\xi$-bordant mod $A$ if and only if they are related by the geometric equivalence relation on the right.

Theorems & Definitions (109)

  • Theorem 1
  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 2
  • Theorem 3
  • Remark 1.4
  • Theorem 1.6
  • Theorem 4
  • Remark 1.7
  • ...and 99 more