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Corks for exotic diffeomorphisms

Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell, Terrin Warren

Abstract

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of $S^2 \times S^2$, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of $S^2 \times S^2$, we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.

Corks for exotic diffeomorphisms

Abstract

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of , is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of , we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.
Paper Structure (25 sections, 31 theorems, 62 equations, 8 figures)

This paper contains 25 sections, 31 theorems, 62 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Pseudo-isotopies and Cerf families of generalized Morse functions
  3. From pseudo-isotopies to Cerf families
  4. Nested eye graphics
  5. Stable diffeomorphisms
  6. The structure of a pseudo-isotopy
  7. The Quinn core
  8. The homotopy type of the Quinn core and the arc condition
  9. A handle decomposition
  10. A cork theorem for 1-stably trivial exotic diffeomorphisms
  11. The sum square move and the associated second homotopy group element
  12. The sum square move
  13. Creating a single circle
  14. An element of the second homotopy group associated to each circle
  15. Many-eyed diffeomorphisms are supported in a null-homotopic homotopy wedge of 2-spheres
  16. ...and 10 more sections

Key Result

Theorem 1.0

Let $X$ be a compact, simply-connected, smooth 4-manifold, and let $f \colon X \rightarrow X$ be a boundary fixing diffeomorphism such that $f$ is 1-stably isotopic to $\operatorname{Id}$. Then there exists a compact, contractible submanifold $C \subseteq X$, and a boundary fixing isotopy of $f$ to

Figures (8)

  • Figure 1: A Cerf graphic for a family in nested eye position. The horizontal direction is the $t$-axis and the vertical direction is the $[0,1]$ direction, recording the critical values of the critical points in the Cerf family.
  • Figure 2: Left: a schematic illustration of $A^t\cup B^t\cup W^t$ for $t$ just before the Whitney move time $5/8$. Right: $A^t\cup B^t\cup \omega^t$ for $t$ right after $5/8$.
  • Figure 3: The regular neighborhoods $\mathcal{N}(A^t\cup B^t\cup W^t)$, $t=5/8-\varepsilon$, and ${\mathcal{N}}(A^t\cup B^t\cup \omega^t)$, $t=5/8+\varepsilon$, are diffeomorphic. In terms of Kirby diagrams, a diffeomorphism is implemented by sliding the $0$-framed $2$-handle corresponding to either $A^t$ or $B^t$ (the left-most and right-most handles respectively) twice over the central $2$-handle corresponding to $W^t$, and then canceling a $1$-, $2$-handle pair.
  • Figure 4: The isotopy is given by $\Phi_t^{-1}$ in the preimage under $q_t$ of the shaded region labeled (i), its reverse in region (iii), and it is constant (as a function of $t$) in region (ii). The isotopy is the identity on the left, top, and bottom boundary arcs of the shaded rectangle (i), i.e. the dashed boundary arcs. The picture is symmetric for $t\leq 1/2$.
  • Figure 5: The sum square move along the sum square $S$ shown in purple.
  • ...and 3 more figures

Theorems & Definitions (74)

  • Theorem 1.0: Diffeomorphism cork theorem
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.3
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • ...and 64 more