Celebratio Mathematica

[1] article M. H. Freedman: “Planes triply tangent to curves with nonvanishing torsion,” Topology 19 : 1 (1980), pp. 1–8. MR 0559472 Zbl 0438.53001

[2] article M. H. Freedman: “A surgery sequence in dimension four; the relations with knot concordance,” Invent. Math. 68 : 2 (1982), pp. 195–226. MR 0666159 Zbl 0504.57016

We present a systematic treatment of the classification problem for compact smooth 4-manifolds \( M \). It is modeled on the surgery exact sequence, the central theorem in the classification of \( n \)-manifolds \( n \geq 5 \). The price for the extension to dimension \( = 4 \) is a hole in \( M \) where a (homotopy) 1-skeleton should be. There is no homotopy theoretic or surgical obstruction to completing \( M \) with a wedge of circles so that the completion has the topology of a compact smooth manifold. This point-set problem is all that stands between 4-manifolds and the tranquility that prevails in higher dimensions.

When \( M \) is simply connected only a point is missing from the model. The applications of this are discussed in [Freedman 1979] and [Freedman and Quinn 1981]. The general theory has application to knot and link theory. In particular, knots with Alexander polynomial \( = 1 \) are characterized geometrically as knots admitting a certain type of “singular slice.” As a lure to low-dimensional topologists, the knot theoretic “applications” are actually presented first as a special case.

This paper was written in 1979. To bring it up-to-date [Freedman 1982] with recent developments one should say that an isolated simply connected end of a smooth 4-manifold is now known to be topologically collared as \( \mathbb{S}^3\times [0,\infty) \). Thus none of the isolated singularities of 4-manifolds contemplated in this paper actually exists in a topological sense. Also the solution to the 4-dimensional Poincaré conjecture identifies a homotopy \( \mathbb{B}^4 \) with \( \mathbb{S}^3 \) boundary as topologically \( \mathbb{B}^4 \). Thus, for example, the untwisted doubles of a knot with Alexander polynomial \( = 1 \) are actually sliced by topologically flat disks in \( \mathbb{B}^4 \). However, the singular nature of the homotopically flat disks and the nonsimply connected ends which we encounter is still an open question.

[3] article M. H. Freedman: “A new technique for the link slice problem,” Invent. Math. 80 : 3 (1985), pp. 453–465. MR 0791669 Zbl 0569.57002

[4] article M. H. Freedman: “A note on topology and magnetic energy in incompressible perfectly conducting fluids,” J. Fluid Mech. 194 (1988), pp. 549–551. MR 0988303 Zbl 0676.76095

[5] article M. H. Freedman and Z.-X. He: “Factoring the logarithmic spiral,” Invent. Math. 92 : 1 (1988), pp. 129–138. MR 0931207 Zbl 0622.30011

[6] article M. H. Freedman and Z.-X. He: “A remark on inherent differentiability,” Proc. Amer. Math. Soc. 104 : 4 (1988), pp. 1305–1310. MR 0937012 Zbl 0689.57021

[7] article M. H. Freedman: “Whitehead\( {}_3 \) is a ‘slice’ link,” Invent. Math. 94 : 1 (1988), pp. 175–182. MR 0958596 Zbl 0678.57002

[8] article M. H. Freedman and Z.-X. He: “Links of tori and the energy of incompressible flows,” Topology 30 : 2 (1991), pp. 283–287. MR 1098922 Zbl 0731.57003

The moduli of curve families have been a useful bridge between analysis and geometric-topological argument. Caratheódory exploited the notion to construct his theory of prime ends. In dimensions three and higher this connection has been extensively developed by Fred Gehring (for example, see [Gehring 1971; 1986]). We continue in this tradition by showing that the naturally defined “conformal moduli” for a disjoint collection of solid tori in \( \mathbb{R}^3 \) cannot all be greater than the constant \( 125\pi/48 \) if the tori are linked in any essential manner. It is natural to conjecture that the optimal lower bound is \( (\sqrt{2}-1)/2\pi \), the modulus of the “solid” Clifford torus.

As an application, we use the topology of linking flow lines to estimate a lower bound on the energy of certain incompressible flows. Roughly, one thinks that an invariant solid torus of spinning fluid may give up energy by elongating like a soda straw, but that this should be prevented if several such tori are linked. To make this precise, an inequality relating modulus and a variant of energy is derived.

[9] article M. H. Freedman and Z.-X. He: “Divergence-free fields: Energy and asymptotic crossing number,” Ann. of Math. (2) 134 : 1 (1991), pp. 189–229. MR 1114611 Zbl 0746.57011

[10] incollection M. H. Freedman and Z.-X. He: “Research announcement on the ‘energy’ of knots,” pp. 219–222 in Topological aspects of the dynamics of fluids and plasmas (University of California at Santa Barbara, August–December 1991). Edited by H. K. Moffatt, G. M. Zaslavasky, P. Comte, and M. Tabor. NATO ASI Series E: Applied Sciences 218. Kluwer Academic Publishers (Dordrecht), 1992. MR 1232232 Zbl 0788.53004

Zheng-Xu He	Related
Pierre Comte	Related
Michael Tabor	Related
Henry Keith Moffatt	Related
George M. Zaslavsky	Related

[11] article S. Bryson, M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius invariance of knot energy,” Bull. Amer. Math. Soc. (N.S.) 28 : 1 (1993), pp. 99–103. MR 1168514 Zbl 0776.57003 ArXiv math/9301212

Zhenghan Wang	Related
Zheng-Xu He	Related
Steve Bryson	Related

[12] article M. H. Freedman: “Link compositions and the topological slice problem,” Topology 32 : 1 (1993), pp. 145–156. MR 1204412 Zbl 0782.57010

[13] article M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius energy of knots and unknots,” Ann. of Math. (2) 139 : 1 (1994), pp. 1–50. MR 1259363 Zbl 0817.57011

Zhenghan Wang	Related
Zheng-Xu He	Related

[14] article M. H. Freedman and F. Luo: “Equivariant isotopy of unknots to round circles,” Topology Appl. 64 : 1 (1995), pp. 59–74. MR 1339758 Zbl 0830.53002

[15] article M. H. Freedman and D. Gabai: “Covering a nontaming knot by the unlink,” Algebr. Geom. Topol. 7 (2007), pp. 1561–1578. MR 2366171 Zbl 1158.57024

Michael H. Freedman

Knots