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 pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 6 jan 2021