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The group of disjoint 2-spheres in 4-space

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Schneiderman,  Rob
Max Planck Institute for Mathematics, Max Planck Society;

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Teichner,  Peter
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1708.00358.pdf
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Citation

Schneiderman, R., & Teichner, P. (2019). The group of disjoint 2-spheres in 4-space. Annals of Mathematics, 190(3), 669-750. doi:10.4007/annals.2019.190.3.1.


Cite as: http://hdl.handle.net/21.11116/0000-0005-52A4-5
Abstract
We compute the group $LM_{2,2}^4$ 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 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 we introduce a number of new 4-dimensional constructions, including maneuvers involving the boundary arcs of Whitney disks.