Celebratio Mathematica — Freedman

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incollection M. H. Freedman: “The disk theorem for four-dimensional manifolds,” pp. 647–663 in Proceedings of the International Congress of Mathematicians (August 16–24, 1983, Warsaw), vol. 1. Edited by Z. Ciesielski and C. Olech. PWN (Warsaw), 1984. MR 0804721 Zbl 0577.57003

The two-dimensional disk \( \mathbb{D}^2 \) seems to serve as a fundamental unit in manifold topology, mediating algebra and geometry. For manifolds of dimension greater than or equal to five, intersection pairings taking values in group rings \( \mathbb{Z}[\pi_1M] \) are crucial to the classification problem. The pairings are translated into precise geometric information by isotopies guided by imbedded two-disks. This is the “Whitney trick” [Whitney 1944], key to both \( s \)-cobordism and (even-dimensional) surgery theorems. The topology of three-dimensional manifolds is closely tied to the fundamental group by the classical disk locating theorems, Dehn’s Lemma and the Loop Theorem. These theorems make the hierarchy theory run and eventually lead to toroidal decomposition. (And conversely, the least understood 3-manifolds are those having no fundamental group to decompose by imbedded disks — homotopy 3-spheres.) One could extend this pattern to dimension two by quoting the continuous-boundary-value Riemann mapping theorem (together with the uniformization theorem) as the 2-dimensional disk theorem.

There is now a 4-dimensional 2-disk imbedding theorem. Its simply connected version was the key to the work on the Poincaré conjecture [Freedman 1982]. The body of this paper is a discussion of its proof, with applications being given at the end.

Czesław Olech	Related
Zbigniew Ciesielski	Related