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The 4-dimensional disc embedding theorem and dual spheres

Mark Powell, Arunima Ray, Peter Teichner

TL;DR

This work fills a foundational gap in the Freedman–Quinn disc embedding program by constructing geometrically dual spheres in the disc embedding theorem for 4-manifolds with good fundamental groups, rather than only algebraic duals. It introduces a refined first-step construction using 1-storey capped towers and Clifford tori to yield many geometrically dual spheres, and proves Freedman–Quinn Proposition 1.6 on generic homotopies of discs and spheres. The paper provides a self-contained enhancement of the disc embedding proof and integrates generic-immersions technology with dual-sphere control to support topological 4-manifold surgery arguments. The resulting framework supports robust surgical manipulations that preserve the fundamental group and enable sphere embedding, hyperbolic decompositions, and related classification results in topological 4-manifolds.

Abstract

We modify the proof of the disc embedding theorem for $4$-manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the statement but not constructed in the proof. We also prove Proposition 1.6 from the Freedman-Quinn book regarding generic homotopies of discs or spheres in a 4-manifolds, which was not proven there.

The 4-dimensional disc embedding theorem and dual spheres

TL;DR

This work fills a foundational gap in the Freedman–Quinn disc embedding program by constructing geometrically dual spheres in the disc embedding theorem for 4-manifolds with good fundamental groups, rather than only algebraic duals. It introduces a refined first-step construction using 1-storey capped towers and Clifford tori to yield many geometrically dual spheres, and proves Freedman–Quinn Proposition 1.6 on generic homotopies of discs and spheres. The paper provides a self-contained enhancement of the disc embedding proof and integrates generic-immersions technology with dual-sphere control to support topological 4-manifold surgery arguments. The resulting framework supports robust surgical manipulations that preserve the fundamental group and enable sphere embedding, hyperbolic decompositions, and related classification results in topological 4-manifolds.

Abstract

We modify the proof of the disc embedding theorem for -manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the statement but not constructed in the proof. We also prove Proposition 1.6 from the Freedman-Quinn book regarding generic homotopies of discs or spheres in a 4-manifolds, which was not proven there.

Paper Structure

This paper contains 11 sections, 11 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The disc embedding theorem
  3. Outline of the proof
  4. Generic immersions and intersection numbers
  5. Applications of geometrically dual spheres in the literature
  6. The sphere embedding theorem
  7. Other instances of geometrically dual spheres
  8. Definitions and operations
  9. Constructing capped gropes and towers with dual spheres
  10. Proof of the disc embedding theorem
  11. Generic immersions in smooth and topological 4-manifolds

Key Result

Theorem A

Let $M$ be a connected $4$-manifold with good fundamental group. Consider a continuous map that is a locally flat embedding on the boundary and that admits algebraically dual spheres $\{g_i\}_{i=1}^k$ satisfying $\lambda(g_i,g_j)=0= \widetilde{\mu}(g_i)$ for all $i,j$. Then there exists a locally flat embedding such that $\overline{F}$ has the same boundary as $F$ and admits a generically immers

Figures (1)

  • Figure 1: Contraction and push-off. Note the intersections of pushed-off surfaces that occur between diagrams one and two and between diagrams four and five in the bottom row of figures, namely one intersection in the past and one intersection in the future between each pair of surfaces pushed off dual caps.

Theorems & Definitions (31)

  • Theorem A: Disc embedding theorem cf. FQ*Theorem 5.1A
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem B: Sphere embedding theorem with framed duals
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Hyperbolic embedding theorem
  • ...and 21 more