Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A 4-dimensional light bulb theorem for disks

Hannah Schwartz

Abstract

We give a 4-dimensional light bulb theorem for properly embedded disks, generalizing recent work of Gabai and Kosanovic-Teichner in certain contexts, and extending the 4-dimensional light bulb theorem for 2-spheres due to Gabai and Schneiderman-Teichner. In particular, we provide conditions under which homotopic disks properly embedded in a compact 4-manifold X with a common dual in the interior of X are smoothly isotopic rel boundary. We also provide a new geometric interpretation of the Dax invariant, to aid in its computation.

A 4-dimensional light bulb theorem for disks

Abstract

We give a 4-dimensional light bulb theorem for properly embedded disks, generalizing recent work of Gabai and Kosanovic-Teichner in certain contexts, and extending the 4-dimensional light bulb theorem for 2-spheres due to Gabai and Schneiderman-Teichner. In particular, we provide conditions under which homotopic disks properly embedded in a compact 4-manifold X with a common dual in the interior of X are smoothly isotopic rel boundary. We also provide a new geometric interpretation of the Dax invariant, to aid in its computation.

Paper Structure

This paper contains 9 sections, 10 theorems, 5 equations, 26 figures.

Table of Contents

  1. Introduction and Motivation
  2. Preliminaries
  3. Regular homotopies
  4. Dual spheres and tubed disks
  5. Self-referential disks and the Dax invariant
  6. Modifying regular homotopies
  7. Self-referential disks with a common dual
  8. Converting homotopies to isotopies
  9. Computing the Dax invariant

Key Result

Lemma \oldthetheorem

Let $D$ be a properly embedded disk in $X$, with dual $G$ in the interior of $X$. For any $g \in \pi_1(X)$ there is a smooth isotopy rel boundary between the self-referential disks $D_g$ and $D_{{\bar{g}}}$ supported away from their common dual $G$.

Figures (26)

  • Figure 1: $3$-dimensional "slices" of the local model of the disk throughout the regular homotopies from Definition \ref{['fwmovedef']}.
  • Figure 2: An immersed disk $\Sigma \subset X$ with two double points and dual $G$ (top), together with a collection $\alpha=\{\alpha_1, \alpha_2\}$ of tubing arcs in $\Sigma$ connecting each double point to the point $\Sigma \pitchfork G$. As in Definition \ref{['tubeddisk']}, this determines an embedded tubed disk $(\Sigma,\alpha)$ (bottom). It is often convenient to picture isotopies between tubed disks as a motion of the pre-images of the tubing arcs in $D^2$ (left).
  • Figure 3: Isotopies of a tubed disk which move only the tubes and leave the rest of the disk fixed, as in Definition \ref{['AB']}. These isotopies, drawn from the perspective of the tube diagram, are identical to those defined in dave:LBT. Re-ordering is also described by Schneiderman-Teichner in Figures $14$ and $15$ of st
  • Figure 4: Three dimensional cross sections of the disks during the tube isotopies A and B from Definition \ref{['AB']} and Figure \ref{['moves2']}: Isotopy A swings the tube $\tau$ across the dual $G$ to a new tube $\tau'$ as indicated by the arrows, and isotopy B moves the tube $\tau$ to a tube $\tau'$ that is nested "inside" of another tube.
  • Figure 5: A descending shadow $(\Sigma, \alpha)$ of the homotopy $h$ in Definition \ref{['shadow']}. By Gabai dave:LBT, the tubed disk $(\Sigma, \alpha)$ is isotopic to $D_0$ (as indicated on the bottom right). The top row shows the disks in the domains of the immersions, decorated by the pre-images of the double points of $\Sigma$, as well as pre-images of the tubing and Whitney arcs.
  • ...and 21 more figures

Theorems & Definitions (32)

  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 22 more