The Group of Disjoint 2-Spheres in 4-Space

TL;DR

This paper computes the link homotopy group of two 2-spheres in 4-space, revealing it as free abelian, and introduces new techniques like Whitney homotopies and algebraic tools to analyze embeddings.

Contribution

It introduces a new Whitney homotopy and applies Freedman's disk theorem to compute the link homotopy group, advancing understanding of 4-dimensional link maps.

Findings

The group of link homotopy classes is free abelian.

Any link map with one embedded component is link homotopic to the unlink.

New 4D constructions and maneuvers involving Whitney disks are developed.

Abstract

We compute the group of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Paul Kirk in this setting. As a corollary, we show that any link map with one topologically embedded component is link homotopic to the unlink. Our proof introduces a new basic link homotopy, which we call a Whitney homotopy, that shrinks an embedded Whitney sphere constructed from four copies of a Whitney disk. Freedman's disk embedding theorem is applied to get the necessary embedded Whitney disks, after constructing sufficiently many accessory spheres as algebraic duals for immersed Whitney disks. To construct these accessory spheres and immersed Whitney disks we use the algebra of metabolic forms over the group ring Z[Z], and introduce a number of new 4-dimensional constructions, including maneuvers involving the boundary arcs of Whitney disks.