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Extendable periodic automorphisms of closed surfaces over the 3-sphere

Chao Wang, Weibiao Wang

Abstract

A periodic automorphism of a surface $Σ$ is said to be extendable over $S^3$ if it extends to a periodic automorphism of the pair $(S^3,Σ)$ for some possible embedding $Σ\to S^3$. We classify and construct all extendable automorphisms of closed surfaces, with orientation-reversing cases included. Moreover, they can all be induced by automorphisms of $S^3$ on Heegaard surfaces. As a by-product, the embeddings of surfaces into lens spaces are discussed.

Extendable periodic automorphisms of closed surfaces over the 3-sphere

Abstract

A periodic automorphism of a surface is said to be extendable over if it extends to a periodic automorphism of the pair for some possible embedding . We classify and construct all extendable automorphisms of closed surfaces, with orientation-reversing cases included. Moreover, they can all be induced by automorphisms of on Heegaard surfaces. As a by-product, the embeddings of surfaces into lens spaces are discussed.

Paper Structure

This paper contains 17 sections, 21 theorems, 106 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classification of periodic maps and main theorems
  3. Basic examples of extendable maps
  4. Type $(+,+)$
  5. Type $(-,-)$
  6. Type $(+,-)$
  7. Type $(-,+)$
  8. Realizations of the extensions
  9. Type $(+,+)$
  10. Type $(-,-)$
  11. Type $(+,-)$
  12. Type $(-,+)$
  13. Necessary conditions for extendability
  14. Type $(-,-)$
  15. Type $(+,-)$
  16. ...and 2 more sections

Key Result

Theorem 1.2

If an automorphism $\phi_0\in{\rm Aut}(S^3)$ induces a periodic map $f\in{\rm Aut}(\Sigma_g)$ with respect to some embedding $\Sigma_g\hookrightarrow S^3$, then $f$ can also be induced by a periodic automorphism $\phi$ of $S^3$ on a Heegaard surface.

Figures (10)

  • Figure 1: $\phi-$invariant graphs ($n=6$).
  • Figure 2: A $\phi-$invariant graph with two components $(n=6)$ and the equidistance surface in a fundamental domain.
  • Figure 3: A genus $1$ Heegaard surface in $S^3$$(d=4)$.
  • Figure 4: A genus $d+1$ Heegaard surface in $S^3$$(d=3)$.
  • Figure 5: Constructing new extendable maps with surgeries ($\tilde{n}=6,\tilde{h}=2,\tilde{s}=6,\tilde{t}=1,\tilde{p}=2,\tilde{q}=3$).
  • ...and 5 more figures

Theorems & Definitions (42)

  • Theorem 1.2
  • Proposition 2.1: Theorem 2.2 in GWWZ and Lemma 4.1 in C1
  • Theorem 2.2: Classification theorem
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • ...and 32 more