nLab 3-manifold

Idea

A 3-manifold is a manifold of dimension 3. (Our default meaning of “manifold” is topological manifold, unless a qualifier is added, e.g., smooth manifold.)

Properties

Triangulability and smoothing

The following is taken from Hatcher:

A pleasant feature of 3-manifolds, in contrast to higher dimensions, is that there is no essential difference between smooth, piecewise linear, and topological manifolds. It was shown by Bing and Moise in the 1950s that every topological 3-manifold can be triangulated as a simplicial complex whose combinatorial type is unique up to subdivision. And every triangulation of a 3-manifold can be taken to be a smooth triangulation in some differential structure on the manifold, unique up to diffeomorphism. Thus every topological 3-manifold has a unique smooth structure, and the classifications up to diffeomorphism and homeomorphism coincide.

Thus it makes no essential difference if we consider 3-manifolds as mere topological manifolds, or as piecewise-linear manifolds or smooth manifolds. It’s often technically convenient to work in the smooth category.

Poincaré conjecture

Theorem

(Poincaré conjecture)

Every simply connected compact 3-manifold without boundary is homeomorphic to the 3-sphere.

Proof

A proof strategy was given by Richard Hamilton: imagine the manifold is equipped with a metric. Follow the Ricci flow of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.

The hard technical part of this program is to show that the passage through the singularities can be controlled. This was finally shown by Grigori Perelman.

Geometrization conjecture

The geometrization conjecture says that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.

Virtually fibered conjecture

The virtually fibered conjecture says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

Examples

• 3-sphere/SU(2)

• $O(3)$ and SO(3)

• lens space

• Hantzsche-Wendt manifold

• Seifert manifold

• hyperbolic 3-manifold

Related concepts

• atoroidal 3-manifold

• Seifert 3-manifold

• hyperbolic 3-manifold

• Dehn surgery

• Kirby calculus

• low dimensional topology

• knot theory

• arithmetic topology

• Chern-Simons theory

• Atiyah 2-framing

• lens space

• associative submanifold

manifolds in low dimension:

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 5-manifolds

• 8-manifolds

References

General

Review:

• William Thurston: Three-dimensional geometry and topology, preliminary draft, University of Minnesota (1992) [1979: ark:/13960/t3714t34v, 1991: pdf, 2002: pdf, pdf]

  the first three chapters of which are published in expanded form as:

• William Thurston: The Geometry and Topology of Three-Manifolds, Princeton University Press (1997) [ISBN:9780691083049, Wikipedia page]

  in particular orbifolds are discussed in chapter 13

• Vladimir Turaev: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, de Gruyter & Co. (1994) [doi:10.1515/9783110435221, pdf]

• Allen Hatcher, The classification of 3-manifolds – a brief overview, (pdf)

• Tomotada Ohtsuki: Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets, World Scientific (2001) [doi:10.1142/4746]

• Bruno Martelli, An Introduction to Geometric Topology (arXiv:1610.02592)

• Neil Strickland, section 12 of: A Bestiary of Topological Objects [pdf, pdf]

The triangulation theorem for 3-manifolds:

• Edwin E. Moise, Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 (doi:10.2307/1969769, jstor:1969769)

3-manifolds as branched covers of the 3-sphere:

• J. Montesinos, A representation of closed orientable 3-manifolds as 3-fold branched coverings of $S^3$, Bull. Amer. Math. Soc. 80 (1974), 845-846 (Euclid:1183535815)

See also

• Raoul Bott, Alberto Cattaneo, Integral Invariants of 3-Manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001)

Hyperbolic 3-manifolds

On hyperbolic 3-manifolds:

• William Thurston, Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds, Annals of Math, 124 (1986), 203–246 (jstor:1971277, arXiv:math/9801019)

• William Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle (arXiv:math/9801045)

• William Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 (euclid.bams/1183548782)

Vafa-Witten theory

Computations of Vafa-Witten invariants of 3-manifolds are given in

• Sergei Gukov, Artan Sheshmani, Shing-Tung Yau, 3-manifolds and Vafa-Witten theory (arXiv:2207.05775).

Quantum invariants

(see also at knot theory)

• Yuya Murakami: A proof of a conjecture of Gukov-Pei-Putrov-Vafa [arXiv:2302.13526]