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

TL;DR

This paper classifies equivariant concordance classes of periodic 2-knots in 4-sphere, introducing a new invariant based on a variation of the Arf invariant, revealing a non-trivial structure unlike the non-equivariant case.

Contribution

It introduces a new invariant for periodic 2-knots that fully classifies them up to equivariant concordance, contrasting with the trivial classification in the non-equivariant case.

Findings

The equivariant concordance group is isomorphic to Z/2Z for all d ≥ 2.

A new invariant fully classifies periodic 2-knots up to equivariant concordance.

Identical classification applies to certain annuli in S^1 × B^3.

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.