Equivariant concordance of periodic 2-knots in S⁴

2025/08/22 by Bohm, Remy
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2508.16778

Abstract

We show that the equivariant concordance group of smooth 2-knots in S⁴ invariant under a ℤ/dℤ action, where the action is given by rotation about an unknotted sphere intersecting the 2-knot in two points, is isomorphic to ℤ/2ℤ for all d ≥ 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 an extension of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds.

Citations

• Equivariant trisections for group actions on four-manifolds
• An equivariant Laudenbach-Poénaru theorem
• Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization
• The equivariant concordance group is not abelian
• A relative version of Rochlin's theorem
• Homotopy versus isotopy: spheres with duals in 4-manifolds
• Embedded Morse theory and relative splitting of cobordisms of manifolds
• Surfaces in 4-manifolds: Concordance, Isotopy, and Surgery
• A survey of group actions on 4-manifolds

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