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$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces

Xiaoming Du, Weibiao Wang

Abstract

For every Sol manifold $M$, we determine the $\mathbb{Z}_2$-Thurston norm of every element in $H_2(M;\mathbb{Z}_2)$. Each Sol manifold is either a torus bundle over the circle or a torus semi-bundle, thus corresponds to a torus map. We discuss the action of this torus map on a curve complex for the torus, whose edges connect curve classes of intersection number 2. For torus bundles over the circle, the $\mathbb{Z}_2$-Thurston norm of any $\mathbb{Z}_2$-homology class equals either zero or the minimum translation distance under the action; and for torus semi-bundles, it equals either zero or the translation distance of a specific curve class. Moreover, we construct incompressible surfaces to realize all the $\mathbb{Z}_2$-homology classes. As a consequence, for any torus bundle over the circle or torus semi-bundle, we determine which non-orientable closed surfaces can be embedded in it.

$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces

Abstract

For every Sol manifold , we determine the -Thurston norm of every element in . Each Sol manifold is either a torus bundle over the circle or a torus semi-bundle, thus corresponds to a torus map. We discuss the action of this torus map on a curve complex for the torus, whose edges connect curve classes of intersection number 2. For torus bundles over the circle, the -Thurston norm of any -homology class equals either zero or the minimum translation distance under the action; and for torus semi-bundles, it equals either zero or the translation distance of a specific curve class. Moreover, we construct incompressible surfaces to realize all the -homology classes. As a consequence, for any torus bundle over the circle or torus semi-bundle, we determine which non-orientable closed surfaces can be embedded in it.
Paper Structure (11 sections, 30 theorems, 37 equations, 8 figures)

This paper contains 11 sections, 30 theorems, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The intersection number 2 curve complex
  3. Incompressible surfaces
  4. Basic concepts
  5. Incompressible surfaces in the thickened torus and the solid torus
  6. Incompressible surfaces in Seifert fibered manifolds
  7. Essential surfaces in the twisted I-bundle over the Klein bottle
  8. Z2-Thurston norm
  9. Torus bundles over S1
  10. Torus semi-bundles
  11. Embeddability of non-orientable closed surfaces

Key Result

Lemma 2.1

The action of ${\rm Aut}(T^2)$ on $\mathcal{C}^{i=2}(T^2)$ is transitive on both the vertex set and the edge set.

Figures (8)

  • Figure 1: $\mathcal{C}^{i=1}(T^2)$: a Farey graph.
  • Figure 2: The three components of $\mathcal{C}^{i=2}(T^2)$.
  • Figure 3: An embedded $\Pi_{1,2}$ in $T^2 \times I$.
  • Figure 4: A fundamental domain for the $\sigma$-action on $S^1\times S^1\times I$.
  • Figure 5: Essential surfaces in a torus semi-bundle.
  • ...and 3 more figures

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3: BW1969
  • Lemma 2.4: Przytycki2019
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2: Przytycki2019
  • ...and 39 more