manifolds and cobordisms
cobordism theory, Introduction
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.)
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.
Every simply connected compact 3-manifold without boundary is homeomorphic to the 3-sphere.
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.
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.
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.
irreducible 3-manifold?
Review:
William Thurston, Geometry and topology of three-manifolds (1980), electronic version 1.1 (2002) available from MSRI (web)
Allen Hatcher, The classification of 3-manifolds – a brief overview, (pdf).
Bruno Martelli, An Introduction to Geometric Topology (arXiv:1610.02592)
The triangulation theorem for 3-manifolds:
3-manifolds as branched covers of the 3-sphere:
See also
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)