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Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

Xiayu Tan

Abstract

For a half-unknotted implanted barbell $β$, we construct two specific pseudo-isotopies, both resulting in that barbell diffeomorphism, and compute the Hatcher-Wagoner invariants for both. We further generalize the results to half-unknotted immersed barbell diffeomorphisms and prove that for every $σ\in π_2 M,γ\in π_1 M$, there is a half-unknotted immersed barbell diffeomorphism $φ_β$ with the second induced Hatcher-Wagoner invariant $Θ(φ_β)=(0,σ)\cdot [γ]$.

Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

Abstract

For a half-unknotted implanted barbell , we construct two specific pseudo-isotopies, both resulting in that barbell diffeomorphism, and compute the Hatcher-Wagoner invariants for both. We further generalize the results to half-unknotted immersed barbell diffeomorphisms and prove that for every , there is a half-unknotted immersed barbell diffeomorphism with the second induced Hatcher-Wagoner invariant .

Paper Structure

This paper contains 6 sections, 19 theorems, 18 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Cerf theory and parameterised handle constructions version
  3. Twin twists and relation with barbell diffeomorphisms
  4. Changing from (1,2)-handle pair to (2,3)-handle pair
  5. Computations for the second Hatcher-Wagoner invariant
  6. Generalizations to immersed barbell diffeomorphisms

Key Result

Theorem 1.1

For a half-unknotted implanted barbell $\beta=(R_0, S, \gamma)$ with $S=\partial \beta_0^\bullet$ where $\beta_0^\bullet: D^3\hookrightarrow M$, by finger-pushing $R_0$ along the arc, we can make $\gamma$ short enough such that $\text{int}(\beta_0^\bullet)\cap \gamma =\emptyset$. Now suppose that $\ Here we identify $\pi_i(M,*_0)$ with $\pi_i(M, \beta_0^\bullet)$ so that $[D_i]\in \pi_2(M,\beta_0^

Figures (15)

  • Figure 1: A general barbell and data needed to compute Hatcher-Wagoner invariants
  • Figure 2: This is a Cerf diagram with 2 eyes, each eye begin from the birth to the end. All births and ends happen in $g^{-1}_t(1/2)$. Except the four birth/death points, every $g_t$ is Morse. For example, $g_{2/3}$ has two non-degenerate critical points of index 2 and 3. There are two crossing points in the diagram. Every arrow denotes a handle slide between critical points with the same index.
  • Figure 3: The implanted barbell $\delta_4$ in $M=S^1\times D^3$
  • Figure 4: Finger-push $R_0$ to get $R$ such that $R\cap S=2$ points
  • Figure 5: Do surgery along $S$ to get an embedded torus $T_R$ in $S^1\times D^3\# S^1\times S^3$ with the standard $S^1$ (green one), this $T_R$ determines an element in $\pi_1(Emb(S^1, S^1\times D^3\# S^1\times S^3),*)$ which is a loop of handle decompositions $\mathcal{H}=\{H_t,t\in S^1\}$ (a loop of (1,2)-handle pair) of $S^1\times D^3\times I$ which results in the barbell diffeomorphism $\delta_4=\tau_W$ where $W=(R,S)$.
  • ...and 10 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 37 more