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

^{3}*M* = \(\mathbb R\)^{3}
# *M*_{1} # *M*_{2} # *M*_{3} # ... ,

where # stands for a connected sum, the initial \(\mathbb R\)^{3} factor
is present for the (topologically) asymptotically flat case, and each *M*_{i} is
a "prime" manifold;
Notice that, for non-orientable *M*_{i}s,
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 *M*_{i} 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 *L*_{p,q}:=
S^{3}/~ ,
where S^{3}= {(*z*_{1},
*z*_{2}) ∈ \(\mathbb C\)^{2} |
||(*z*_{1},
*z*_{2})|| = 1}, (*z*_{1},
*z*_{2}) ~ (*ω* *z*_{1},
*ω*^{q}* z*_{2}),
*ω*:= exp{2πi/*p*}, and *p* and *q* are relatively
prime integers; S^{3} is a *p*:1 cover; __ Example__:
*L*_{2,1} = 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 \(\mathbb R\)P^{3}];
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 *β*_{1}(*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. 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].

@ __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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 4
feb 2017