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Equivariant concordance of periodic 2-knots in $S^4$

Remy Bohm

Abstract

We show that the smooth equivariant concordance group of 2-knots in $S^4$ invariant under a linear $\mathbb{Z}/d\mathbb{Z}$ action is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ for all $d \geq 2$. This is in contrast to the non-equivariant case, in which all 2-knots are slice. We construct a new invariant for these 2-knots, which we call periodic, and show that it fully classifies them up to equivariant concordance. The invariant depends on a variation of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds with respect to a chosen spin structure. Our proof also shows an identical classification for certain annuli in $S^1 \times B^3$ up to concordance rel. boundary.

Equivariant concordance of periodic 2-knots in $S^4$

Abstract

We show that the smooth equivariant concordance group of 2-knots in invariant under a linear action is isomorphic to for all . This is in contrast to the non-equivariant case, in which all 2-knots are slice. We construct a new invariant for these 2-knots, which we call periodic, and show that it fully classifies them up to equivariant concordance. The invariant depends on a variation of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds with respect to a chosen spin structure. Our proof also shows an identical classification for certain annuli in up to concordance rel. boundary.

Paper Structure

This paper contains 33 sections, 33 theorems, 28 equations, 14 figures.

Table of Contents

  1. Preliminaries on annuli and periodic 2-knots
  2. Periodic 2-knots
  3. Montesinos annuli
  4. Concordance group structure
  5. Concordance of annuli
  6. Equivariant concordance
  7. An extension of the Arf invariant
  8. Background
  9. The Arf invariant of a null-homologous knot in a spin 3-manifold
  10. The induced characterization from a spin structure
  11. Definition of the invariant
  12. Spin characteristic fillings
  13. The corresponding smooth concordance problem
  14. Extension to arcs and additivity
  15. Embedded Morse theory and codimension 2 intersections
  16. ...and 18 more sections

Key Result

Theorem \ref{thm: equi conc}

For each positive integer $d$, there exists a $\mathbb{Z}/2\mathbb{Z}$-valued, additive concordance invariant $\alpha_d$ for $d$-periodic 2-knots $\tilde{S}$ such that: Thus the smooth equivariant concordance group $\mathcal{C}^{d} _2$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ for all $d$.

Figures (14)

  • Figure 1: Two projections of an unknotted Montesinos annulus and its removed sphere. On the left, we see the annulus $N$ with its boundary components on $\partial (\nu(R))$. On the right and in a different 3-dimensional projection, we see $R$ locally as a plane, and one slice of the corresponding sphere $S$ for $N$ as a circle intersecting $R$ in two points.
  • Figure 2: A movie of the one-twist spun trefoil, illustrated by twisting an the spin axis $A$ on the boundary while fixing the trefoil tangle $T$. The Seifert surfaces shown patch together to form a 3-ball in which the complement of $A$ is fibered by punctured tori with the trefoil monodromy.
  • Figure 3: A schematic of one annulus stacked on top of another.
  • Figure 4: A band pass achieved by 0-framed 2-handles.
  • Figure 5: A trefoil in punctured $S^2 \times S^2$ which bounds a characteristic disk with relative Euler number 8.
  • ...and 9 more figures

Theorems & Definitions (94)

  • Theorem \ref{thm: equi conc}
  • Theorem \ref{thm: annuli classification}
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • ...and 84 more