3-Manifolds| 3-Manifolds |
In General
* History: In the late 1970s,
William Thurston revolutionized our understanding of 3-manifolds; He stated a
far-reaching geometrization conjecture, proved it for a large class of manifolds,
called Haken manifolds, and posed 24 open problems describing his vision of the
structure of 3-manifolds; By 2012, The program had been essentially completed,
including Grigori Perelman's 2002 proof of the geometrization conjecture; 2002,
Ian Agol's proof of "Wise's conjecture" settled the last four of Thurston's
questions in one stroke.
* Topological classification:
A full one has not been found yet, but various different decompositions are
possible, the prime-decomposition, the torus decomposition and the Heegard
decomposition.
* Prime decomposition:
Any three-manifold can be decomposed in an essentially unique way as
3M = \(\mathbb R\)3 # M1 # M2 # M3 # ... ,
where # stands for a connected sum, the initial \(\mathbb R\)3 factor
is present for the (topologically) asymptotically flat case, and each Mi is
a "prime" manifold;
Notice that, for non-orientable Mis,
the connected sum # does not specify in which of the two inequivalent ways
the operation is performed (for the orientable ones, there is only one orientation-preserving
possibility); The irreducible pieces Mi have
not been classified.
* Cobordism: All closed 3-manifolds are in the same cobordism class.
* Differentiable structure:
All closed 3-manifolds have a unique differentiable structure.
* Spin structure: All 3-manifolds admit a spin structure.
* Decidability: The set of compact 3-manifolds is algorithmically decidable,
i.e., it has an algorithmic description.
> Online resources: see Wikipedia page.
Examples > s.a. laplace operator.
* Lens spaces: 3D constant
positive-curvature manifolds, obtained as quotients of the three-sphere; Denoted
by Lp,q:= S3/~ ,
where S3= {(z1,
z2) ∈ \(\mathbb C\)2 |
||(z1,
z2)|| = 1}, (z1,
z2) ~ (ω z1,
ωq z2),
ω:= exp{2πi/p}, and p and q are relatively
prime integers; S3 is a p:1 cover; Example:
L2,1 = SO(3).
@ Lens spaces: Bellon CQG(06) [harmonics
from holonomies, and cosmology]; > s.a. knots.
@ Non-orientable: Amendola & Martelli T&A(03),
T&A(05)
[small complexity]; Casali T&A(04) [complexity].
@ Compact hyperbolic: Culler et al Top(98) [smallest]; Kramer ap/04 [group
actions and symmetries].
@ Other: in Freed & Gompf PRL(91)
[Brieskorn sphere]; Scannell CQG(01)
[spacelike slices of flat spacetimes];
Bray & Neves AM(04)
[prime, Yamabe invariant greater than that of \(\mathbb R\)P3];
Boileau & Weidmann Top(05) [with 2-generated fundamental group].
Invariants > s.a. knot invariants.
* Complexity: The numbers of manifolds
of complexity 0, 1, 2, 3, 4, 5, 6, respectively, is 3, 2, 4, 7, 14, 31, 74;
The first hyperbolic 3-manifold occurs at complexity 9.
* Volume: For hyperbolic ones,
with curvature normalized to –1; Problem: For each V, there
are finitely many 3-manifolds with volume < V; Which is the smallest?
* Other: Reshetikhin-Turaev
invariant; > s.a.
topological field theories including chern-simons
theory; spin networks.
@ General references: Stewart Nat(89)mar [volume];
Reshetikhin & Turaev IM(91) [from links and quantum groups];
Kauffman & Lins 94;
Bott & Cattaneo JDG(98)dg/97,
JDG(99)m.GT/98 [integral];
Liu Top(99);
Korepanov JNMP(01)m.GT/00 [PL];
Ohtsuki 01; Turaev 01, 03 [torsion];
Korepanov & Martyushev JNMP(02); McDuff BAMS(06)
[and Floer theory, Ozsváth-Szabó]; Prasolov & Sossinsky 96 [knots, links, braids]; Wintraecken & Vegter T&IA(13).
@ Turaev-Viro: King T&A(07)
[ideal]; Alagic et al PRA(10) [approximation as a universal problem for quantum computation]; > s.a. Turaev-Viro
Theory.
@ And topological quantum field theory: Bakalarska & Broda AIP(98)ht/99,
FdP(00)ht/99;
Ramadevi & Naik CMP(00)
[Lickorish
invariant and Chern-Simons theory]; Kaul & Ramadevi CMP(01)ht/00 [from
Chern-Simons theory]; Garnerone et al ATMP(09)-qp/07 [in
SU(2) Chern-Simons-Witten topological
quantum field theory]; Miković & Martins a1002-conf
[from Chern-Simons theory and spin foams].
@ Finite-type: Garoufalidis et al G&T(01)m.GT/00 [and
trivalent graphs].
@ Relationships: Guadagnini & Pilo CMP(98); Mariño & Moore
NPB(99) [and 4D Donaldson-Witten invariants].
@ With boundary: Murakami & Ohtsuki CMP(97) [from universal quantum
invariant].
@ Classification: Milnor AJM(62); Hendricks BAMS(77);
Thompson BAMS(98)
[algorithmic]; Morgan BAMS(05); Matveev 10 [and algorithmic topology].
@ Related topics: Baseilhac & Benedetti m.GT/01;
Harvey G&T(02)
[cut number not bounded below by β1(X)/3],
Top(05)
[from fundamental group]; Ozsváth & Szabó AM(04)
[and holomorphic disks]; Cavicchioli & Spaggiari DM(08) [genus-2, representation
by family of integers].
Other Structure and Concepts > s.a. affine connections;
diffeomorphisms [including Smale conjecture]; 3D geometry.
@ Surfaces, submanifolds: Carter 95 [surfaces in 3-space];
Gluck & Pan Top(98) [embedded 2-surfaces];
Camacho & Camacho T&A(07) [codimension-1 foliations];
Fernández & Mira DG&A(07),
Torralbo DG&A(10) [constant-mean-curvature surfaces in homogeneous 3-manifolds].
@ Hyperbolic: Fenley Top(98) [foliations];
Gabai et al AM(03) [homotopy hyperbolic].
@ Related topics: Neumann & Swarup G&T(97)
[decompositions]; Carter 11 [diagrammatic algebra]; > s.a. embeddings; foliations; knots.
References > s.a. conjectures [Smith]; topological field theory.
@ Simple, or for physicists: Thurston & Weeks SA(84)jul;
Giulini IJTP(94)gq/93;
Thurston CQG(98); news Quanta(12)oct [readable review].
@ Topology: Neuwirth 75;
Hempel 76;
Thurston 78;
Jaco 80; Brown &
Thickstun ed-82;
Bing 83;
Fenn ed-85;
Thurston 97;
Vigara & Lozano-Rojo 16 [filling Dehn surfaces]; Cui et al a2101 [geometric topology and quantum topology].
@ And computers: Lins 95 [gems].
@ Homeomorphisms: Cesar de Sa & Rourke BAMS(79).
@ Surgery / Links: in Rolfsen 76, ch9;
Kirby IM(78).
@ Framings: Atiyah Top(90).
@ Related topics: Schoen & Yau PNAS(78),
AM(79), PRL(79) & refs;
Hendriks & Laudenbach Top(84);
Friedman & Witt Top(86);
Freedman & Feng 89;
Gabai & Oertel AM(89) [laminations];
Crane CMP(91);
Kwasik & Rosicki T&IA(10) [stabilization];
Stillwell BAMS(12) [Poincaré and early history];
Bestvina BAMS(14) [Thurston's vision].
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 6 jan 2021