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Flexible Surfaces in $\mathbb{C}P^2$ and $S^2\times S^2$

Joshua Lehman

Abstract

A surface $Σ$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,Σ) \to (M,Σ)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$ within any prescribed non-characteristic homology class. Within characteristic homology classes there is a spin structure obstructing flexibility and we construct so-called spin-flexible representatives.

Flexible Surfaces in $\mathbb{C}P^2$ and $S^2\times S^2$

Abstract

A surface in a 4-manifold is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism . We construct flexible surfaces in and within any prescribed non-characteristic homology class. Within characteristic homology classes there is a spin structure obstructing flexibility and we construct so-called spin-flexible representatives.
Paper Structure (11 sections, 29 theorems, 64 equations, 9 figures)

This paper contains 11 sections, 29 theorems, 64 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Extension Methods and Rochlin's Quadratic Form
  3. Rochlin's quadratic form.
  4. Extension methods.
  5. Framed and Spin mapping class groups
  6. Framings
  7. $\mathbb{Z}/r\mathbb{Z}$ winding number functions.
  8. (Spin) Flexible Surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$
  9. An integral lift of the linking form on a stabilization of $F_\beta$.
  10. Surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$
  11. A Single stabilization for smooth plane curves

Key Result

Theorem A

Let $M = \mathbb{C}P^2$ or $S^2 \times S^2$. Fix $x \in H_2(M;\mathbb{Z})$. If $x$ is characteristic, there exists a spin-flexible representative for $x$, otherwise there is a flexible representative for $x$.

Figures (9)

  • Figure 1: A well-known isotopy of a standard annulus in $S^3 \times I$ rel boundary inducing the square of a (right handed) twist along a core curve.
  • Figure 2: Depicted is an axis $\alpha \subset S^3$ for a 2-component unlink $\gamma_1\cup \gamma_2$. The unlink is transverse to the pages of the open book decomposition $\pi : S^3\backslash \alpha \to S^1$.
  • Figure 3: The union $a_{k,L}\cup a_{k,R} \cup \Delta$ forms a 3-component unlink in $S^3$, similarly for $b_{k,L} \cup b_{k,R} \cup \Delta$.
  • Figure 4: The framed Seifert surface $F_\beta$ associated to the word $\beta = (\sigma_1\sigma_2^{-1})^3$ together with its braid word assemblage of Hopf band curves.
  • Figure 5: We stabilize $F_\beta$ along the chain of curves $\{\mu,c_1,\ldots,c_{2g-1}\}$.
  • ...and 4 more figures

Theorems & Definitions (65)

  • Theorem A
  • Theorem B
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 55 more