Celebratio Mathematica

article M. H. Freedman: “The topology of four-dimensional manifolds,” J. Differential Geom. 17 : 3 (1982), pp. 357–453. MR 0679066 Zbl 0528.57011

Manifold topology enjoyed a golden age in the late 1950s and 1960s. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained.

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

incollection M. H. Freedman: “A geometric reformulation of 4-dimensional surgery,” pp. 133–141 in Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3 (1986). MR 0872483 Zbl 0898.57005

incollection M. H. Freedman: “Are the Borromean rings \( A \)-\( B \)-slice?,” pp. 143–145 in Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3. Elsevier Science B.V. (North-Holland) (Amsterdam), 1986. MR 0872484 Zbl 0627.57004

article M. H. Freedman and X.-S. Lin: “On the \( (A,B) \)-slice problem,” Topology 28 : 1 (1989), pp. 91–110. MR 0991101 Zbl 0845.57016

article M. H. Freedman and P. Teichner: “4-manifold topology I: Subexponential groups,” Invent. Math. 122 : 3 (1995), pp. 509–529. MR 1359602 Zbl 0857.57017

The technical lemma underlying the 5-dimensional topological \( s \)-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating \( \pi_1 \)-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the \( \pi_1 \)-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in thisclass are amenable and no amenable group is known to lie outside this class.

article M. Freedman and V. Krushkal: “Topological arbiters,” J. Topol. 5 : 1 (February 2010), pp. 226–247. ArXiv 1002.1063

This paper initiates the study of topological arbiters, a concept rooted in Poincaré–Lefschetz duality. Given an \( n \)-dimensional manifold \( W \), a topological arbiter associates a value 0 or 1 to codimension-0 submanifolds of \( W \), subject to natural topological and duality axioms. For example, there is a unique arbiter on \( \mathbb{R}P^2 \), which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredients underlying the 4-dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in \( \mathbb{S}^3 \) the construction relies on the existence of nilpotent embedding obstructions in dimension 4, reflected in particular by the Milnor group. In higher dimensions novel arbiters are produced using non-trivial squares in stable homotopy theory.

The concept of “topological arbiter” derives from percolation and from 4-dimensional surgery. It is not the purpose of this paper to advance either of these subjects, but rather to study the concept for its own sake. However, in the appendices we give both an application to percolation, and the current understanding of the relationship between arbiters and surgery. An appendix also introduces a more general notion of a multi-arbiter. Properties and applications are discussed, including a construction of non-homological multi-arbiters.

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Michael H. Freedman

The A-B slice problem

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