Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on three-fold branched covers of $S^4$

Ryan Blair, Patricia Cahn, Alexandra Kjuchukova, Jeffrey Meier

Abstract

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property.

A note on three-fold branched covers of $S^4$

Abstract

We show that any 4-manifold admitting a -trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in , smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property.

Paper Structure

This paper contains 10 sections, 6 theorems, 9 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Trisections
  4. Singular bridge trisections and branched covers of $S^4$
  5. Fox colorings and Hilden's map
  6. Proof of the main theorem
  7. Revisiting Hilden's result
  8. Normalized bridge splittings and Piergallini moves
  9. Proof of Theorem \ref{['thmx:main']}
  10. Examples

Key Result

Theorem A

Let $X$ be a 4--manifold that admits a $(g;k_1,k_2,0)$--trisection $\mathbb T$. Then $\mathbb T$ is an irregular 3--fold cover of the standard trisection $\mathbb{T}_0$ of $S^4$. The branching set $\mathcal{S}$ is a closed, connected surface, which is smoothly embedded in $S^4$ away from one singula

Figures (5)

  • Figure 1: A 3--coloring of $(k+2)$-component unlink $L$ in $(g+2)$--bridge position, with $g>k$.
  • Figure 2: Hilden's 3--fold irregular cover of $S^2$ by a surface $\Sigma_3$ of genus 3. $\Sigma_3$ is a $\mathbb{Z}/2\mathbb{Z}$ quotient of a 6--fold regular dihedral cover $\Sigma_{10}$ of $S^2$.
  • Figure 3: A normalized link diagram. $B$ is a 3--colored braid.
  • Figure 4: A $b$--bridge splitting of an unlink, together with red arcs illustrating a pairing of the boundary points whose union with either tangle is connected. The boxed portion of the figure is given by (a) or (b) based on whether the number of maxima in the leftmost component of the unlink is odd or even, respectively.
  • Figure 5: Tricolored tri-plane diagrams of some irregular 3--fold covers of $S^4$. Below each diagram is the total space of the corresponding cover; the homeomorphism type of the branching set; the singularities of the embedding.

Theorems & Definitions (18)

  • Theorem A
  • Corollary B
  • Remark 1.1
  • proof : Proof of Corollary \ref{['corox:main']}
  • Corollary C
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 8 more