article
M. H. Freedman :
“Planes triply tangent to curves with nonvanishing torsion Topology
19 : 1
(1980 ),
pp. 1–8 .
MR
0559472 Zbl
0438.53001
Abstract
BibTeX
Experimentation with a closed loop of wire and a desk top quickly leads to the conclusion that except for certain special configurations, only a finite number of planes are tangent to a given curve \( \alpha(t) \) at three places. The main result is that generically this number is even when the torsion \( \tau_{\alpha}(t) \) is nonvanishing.
@article {key0559472m,
AUTHOR = {Freedman, Michael H.},
TITLE = {Planes triply tangent to curves with
nonvanishing torsion},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {19},
NUMBER = {1},
YEAR = {1980},
PAGES = {1--8},
DOI = {10.1016/0040-9383(80)90027-0},
NOTE = {MR:0559472. Zbl:0438.53001.},
ISSN = {0040-9383},
}
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
Abstract
BibTeX
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.
@article {key0666159m,
AUTHOR = {Freedman, Michael H.},
TITLE = {A surgery sequence in dimension four;
the relations with knot concordance},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {68},
NUMBER = {2},
YEAR = {1982},
PAGES = {195--226},
DOI = {10.1007/BF01394055},
NOTE = {MR:0666159. Zbl:0504.57016.},
ISSN = {0020-9910},
}
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
BibTeX
@article {key0791669m,
AUTHOR = {Freedman, Michael H.},
TITLE = {A new technique for the link slice problem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {80},
NUMBER = {3},
YEAR = {1985},
PAGES = {453--465},
DOI = {10.1007/BF01388725},
NOTE = {MR:0791669. Zbl:0569.57002.},
ISSN = {0020-9910},
}
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
BibTeX
@article {key0988303m,
AUTHOR = {Freedman, Michael H.},
TITLE = {A note on topology and magnetic energy
in incompressible perfectly conducting
fluids},
JOURNAL = {J. Fluid Mech.},
FJOURNAL = {Journal of Fluid Mechanics},
VOLUME = {194},
YEAR = {1988},
PAGES = {549--551},
DOI = {10.1017/S002211208800309X},
NOTE = {MR:0988303. Zbl:0676.76095.},
ISSN = {0022-1120},
}
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
People
BibTeX
@article {key0931207m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
TITLE = {Factoring the logarithmic spiral},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {92},
NUMBER = {1},
YEAR = {1988},
PAGES = {129--138},
DOI = {10.1007/BF01393995},
NOTE = {MR:0931207. Zbl:0622.30011.},
ISSN = {0020-9910},
}
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
Abstract
People
BibTeX
@article {key0937012m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
TITLE = {A remark on inherent differentiability},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {104},
NUMBER = {4},
YEAR = {1988},
PAGES = {1305--1310},
DOI = {10.2307/2047633},
NOTE = {MR:0937012. Zbl:0689.57021.},
ISSN = {0002-9939},
}
article
M. H. Freedman :
“Whitehead\( {}_3 \) is a ‘slice’ link Invent. Math.
94 : 1
(1988 ),
pp. 175–182 .
MR
0958596 Zbl
0678.57002
Abstract
BibTeX
We show that the Whitehead doublet \( \mathrm{Wh}(L) \) of a two component link \( L \) is (topologically flat) slice if and only if the linking number of \( L \) is zero. When they exist, the slices may be chosen so that the complement (\( \mathbb{B}^4-\text{slices} \) ) is homotopy equivalent to a wedge of two circles, \( \mathbb{S}^1\vee \mathbb{S}^1 \) , with certain meridinal loops of \( \mathrm{Wh}(L) \) freely generating \( \pi_1(\mathbb{B}^4-\text{slices}) \) .
@article {key0958596m,
AUTHOR = {Freedman, Michael H.},
TITLE = {Whitehead\${}_3\$ is a ``slice'' link},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {94},
NUMBER = {1},
YEAR = {1988},
PAGES = {175--182},
DOI = {10.1007/BF01394351},
NOTE = {MR:0958596. Zbl:0678.57002.},
ISSN = {0020-9910},
}
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
Abstract
People
BibTeX
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.
@article {key1098922m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
TITLE = {Links of tori and the energy of incompressible
flows},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {30},
NUMBER = {2},
YEAR = {1991},
PAGES = {283--287},
DOI = {10.1016/0040-9383(91)90014-U},
NOTE = {MR:1098922. Zbl:0731.57003.},
ISSN = {0040-9383},
}
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
People
BibTeX
@article {key1114611m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
TITLE = {Divergence-free fields: {E}nergy and
asymptotic crossing number},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {134},
NUMBER = {1},
YEAR = {1991},
PAGES = {189--229},
DOI = {10.2307/2944336},
NOTE = {MR:1114611. Zbl:0746.57011.},
ISSN = {0003-486X},
}
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
People
BibTeX
@incollection {key1232232m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
TITLE = {Research announcement on the ``energy''
of knots},
BOOKTITLE = {Topological aspects of the dynamics
of fluids and plasmas},
EDITOR = {Moffatt, H. K. and Zaslavasky, G. M.
and Comte, P. and Tabor, M.},
SERIES = {NATO ASI Series E: Applied Sciences},
NUMBER = {218},
PUBLISHER = {Kluwer Academic Publishers},
ADDRESS = {Dordrecht},
YEAR = {1992},
PAGES = {219--222},
NOTE = {(University of California at Santa Barbara,
August--December 1991). MR:1232232.
Zbl:0788.53004.},
ISBN = {9780792319009},
}
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
Abstract
People
BibTeX
A physically natural potential energy for simple closed curves in \( \mathbb{R}^3 \) is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant \( M \) is estimated.
@article {key1168514m,
AUTHOR = {Bryson, Steve and Freedman, Michael
H. and He, Zheng-Xu and Wang, Zhenghan},
TITLE = {M{\"o}bius invariance of knot energy},
JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {28},
NUMBER = {1},
YEAR = {1993},
PAGES = {99--103},
DOI = {10.1090/S0273-0979-1993-00348-3},
NOTE = {ArXiv:math/9301212. MR:1168514. Zbl:0776.57003.},
ISSN = {0273-0979},
}
article
M. H. Freedman :
“Link compositions and the topological slice problem Topology
32 : 1
(1993 ),
pp. 145–156 .
MR
1204412 Zbl
0782.57010
BibTeX
@article {key1204412m,
AUTHOR = {Freedman, Michael H.},
TITLE = {Link compositions and the topological
slice problem},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {32},
NUMBER = {1},
YEAR = {1993},
PAGES = {145--156},
DOI = {10.1016/0040-9383(93)90043-U},
NOTE = {MR:1204412. Zbl:0782.57010.},
ISSN = {0040-9383},
}
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
People
BibTeX
@article {key1259363m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu
and Wang, Zhenghan},
TITLE = {M\"obius energy of knots and unknots},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {139},
NUMBER = {1},
YEAR = {1994},
PAGES = {1--50},
DOI = {10.2307/2946626},
NOTE = {MR:1259363. Zbl:0817.57011.},
ISSN = {0003-486X},
}
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
Abstract
People
BibTeX
Suppose that \( \gamma_0 \) is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup \( G \) of the Möbius group of \( S^3 = \) the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy \( \gamma_t \) , \( 0\leq t\leq 1 \) , from \( \gamma_0 \) to a round circle \( \gamma_1 \) .
@article {key1339758m,
AUTHOR = {Freedman, Michael H. and Luo, Feng},
TITLE = {Equivariant isotopy of unknots to round
circles},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {64},
NUMBER = {1},
YEAR = {1995},
PAGES = {59--74},
DOI = {10.1016/0166-8641(94)00086-I},
NOTE = {MR:1339758. Zbl:0830.53002.},
ISSN = {0166-8641},
}
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
Abstract
People
BibTeX
There exists an open 3-manifold \( M \) and a simple closed curve \( \gamma \subset M \) such that \( \pi_1(M\backslash\gamma) \) is infinitely generated, \( \pi_1(M) \) is finitely generated and the preimage of \( \gamma \) in the universal covering of \( M \) is equivalent to the standard locally finite set of vertical lines in \( \mathbb{R}^3 \) , that is, the trivial link of infinitely many components in \( \mathbb{R}^3 \) .
@article {key2366171m,
AUTHOR = {Freedman, Michael H. and Gabai, David},
TITLE = {Covering a nontaming knot by the unlink},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {7},
YEAR = {2007},
PAGES = {1561--1578},
DOI = {10.2140/agt.2007.7.1561},
NOTE = {MR:2366171. Zbl:1158.57024.},
ISSN = {1472-2747},
}
