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Equivariant Heegaard genus of reducible 3-manifolds

Scott A. Taylor

Abstract

The equivariant Heegaard genus of a 3-manifold $M$ with the action of a finite group $G$ of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for $M$. Although a Heegaard splitting of a reducible manifold is reducible and although if $M$ is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number.

Equivariant Heegaard genus of reducible 3-manifolds

Abstract

The equivariant Heegaard genus of a 3-manifold with the action of a finite group of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for . Although a Heegaard splitting of a reducible manifold is reducible and although if is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number.

Paper Structure

This paper contains 14 sections, 29 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General Notions, Orbifold Sums, and Orbifold Heegaard surfaces
  3. Factorizations
  4. Orbifold Heegaard Splittings
  5. Examples of additivity and sub-additivity
  6. Orbifold thin position
  7. Lower bounds
  8. Analyzing orbifold compressionbodies
  9. Equivariant Heegaard genus and the order of the group
  10. Comparatively small factors
  11. Upper Bounds
  12. Examples of super additivity
  13. A general upper bound
  14. Efficient Factorizations

Key Result

Theorem 1

Equivariant Heegaard genus can be sub-additive, additive, or super-additive. In particular,

Figures (6)

  • Figure 1: An example of a vp-compressionbody $(C, T_C)$. It has one ghost arc, one core loop, one bridge arc, and three vertical arcs. The horizontal lines represent a closed, possibly disconnected surface $F$.
  • Figure 2: The two types of pillow.
  • Figure 3: An example of an orbifold with underlying 3-manifold $S^3$. The thick circles represent thick spheres and the thin circle is a thin sphere of a multiple vp-bridge surface $\mathcal{H}$. Arbitrary gluing maps preserving the punctures pointwise can be used along the thick spheres. For $a \in \mathbb N^\infty_2$ we have $\operatorname{net}x_\omega(\mathcal{H}) = \frac{1}{6} + \frac{1}{a}$ and the orbifold characteristic of the thin sphere is $\frac{1}{6} - \frac{1}{a}$. Thus, for $a \geq 6$, $\mathcal{H}^-$ does not contain a spherical orbifold. As $a \to \infty$, we approach 1/6.
  • Figure 4: The covering of the orbifold $(M,T)$ by $(W, \varnothing)$ in Example \ref{['ex 1']}. The manifold $M$ is a lens space and $W = M \times M$. The surface $S$ is the union of two disjoint spheres; it is a double cover of the unpunctured sphere $\overline{S}$. The knot $T$ is an unknot contained in 3-ball bounded by the sphere $S$. The horizontal lines represent bridge surfaces for $(M,T)$ and $(W, \varnothing)$, with $\overline{H}$ being a twice punctured torus and $H$ being a genus 2 surface.
  • Figure 5: The singular set $T_1$ and the bridge sphere $H_1$ for the orbifold $(M_1, T_1)$ in Example \ref{['exam 2']}
  • ...and 1 more figures

Theorems & Definitions (66)

  • Theorem
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: c.f. Petronio
  • Theorem 2.4: after Petronio, Hog-Angeloni--Matveev
  • Theorem 2.5: Equivariant Sphere Theorem MY-EST (c.f. Dunwoody)
  • Definition 2.6: Handle Structures
  • Lemma 2.7: after Zimmermann Zimmermann96
  • proof
  • ...and 56 more