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A knot-theoretic tour of dimension four

Márton Beke, Kyle Hayden

TL;DR

These notes survey how knot theory illuminates 4-manifold topology through handle decompositions and the intersection form $Q_X$, illustrating the read-off of $\pi_1$ and $H_2$ from 1- and 2-handles and the presentation of $Q_X$ via linking matrices. They integrate deep obstructions—Rohlin’s invariant, Freedman’s topological results, and E8 plumbing—showing when smooth structures differ from topological ones, and highlight the Thom conjecture and its symplectic generalizations to constrain minimal genus. The local sliceness problem via the slice genus $g_4(K)$ and trace embedding lemmas connects knot theory to 4-manifold embeddings and obstructions like Alexander polynomials and $s$-invariants. Finally, the notes relate contact and Stein geometry to smooth topology, using Legendrian invariants and Stein handle theory to derive adjunction-type inequalities and genus bounds that bridge complex, symplectic, and smooth perspectives on 4-manifolds.

Abstract

These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the Rényi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.

A knot-theoretic tour of dimension four

TL;DR

These notes survey how knot theory illuminates 4-manifold topology through handle decompositions and the intersection form , illustrating the read-off of and from 1- and 2-handles and the presentation of via linking matrices. They integrate deep obstructions—Rohlin’s invariant, Freedman’s topological results, and E8 plumbing—showing when smooth structures differ from topological ones, and highlight the Thom conjecture and its symplectic generalizations to constrain minimal genus. The local sliceness problem via the slice genus and trace embedding lemmas connects knot theory to 4-manifold embeddings and obstructions like Alexander polynomials and -invariants. Finally, the notes relate contact and Stein geometry to smooth topology, using Legendrian invariants and Stein handle theory to derive adjunction-type inequalities and genus bounds that bridge complex, symplectic, and smooth perspectives on 4-manifolds.

Abstract

These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the Rényi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.
Paper Structure (13 sections, 26 theorems, 25 equations, 21 figures)

This paper contains 13 sections, 26 theorems, 25 equations, 21 figures.

Table of Contents

  1. Handles, homotopy, and homology
  2. The intersection form
  3. Linking numbers
  4. Calculating intersection forms from handle diagrams
  5. Basic properties for intersection forms
  6. Constraints on intersection forms
  7. Addendum: Rohlin's invariant and beyond
  8. The minimal genus problem
  9. The local minimal genus problem
  10. An exotic $\boldsymbol{\mathbf{R}^4}$
  11. Addendum and additional exercises
  12. Contact geometry and the slice-Bennequin inequality
  13. Stein surfaces

Key Result

Theorem 1

There exists a closed, simply connected, topological 4-manifold that does not admit a smooth structure.

Figures (21)

  • Figure 1: Exhibiting a singular 2-sphere inside a 4-manifold built from one 0-handle and one 2-handle, known as a knot trace (cf. Example \ref{['ex:wedge']}).
  • Figure 2: Kirby diagrams for a 4-manifold that can be expressed as the 0-trace of two different knots.
  • Figure 3: Handle diagrams for the 4-manifolds in Examples \ref{['ex:running-ex']}, \ref{['ex:1-handle-examples']}, and \ref{['ex:H2']}.
  • Figure 4: The two spheres are identified, and the other part of the circle travels through the core of the handle which we don't picture.
  • Figure 5: Modifying an attaching curve by a homotopy that involves a crossing change.
  • ...and 16 more figures

Theorems & Definitions (74)

  • Theorem : Freedman freedman
  • Theorem : Donaldson donaldson, Freedman freedman
  • Theorem 1.1: Freedman freedman
  • Remark 1.2: Anatomy of a handle
  • Example 1.3
  • Example 1.5
  • Example 1.6
  • Example 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 64 more