What are the different ways of defining 3-manifolds?

I wonder what are the different ways of defining the 3-manifold. Obviously for average human being it is difficult to imagine the 3-manifold. Therefore the presentation or visualisation of such object is important to understand what we are talking about.

Here are the possible ways which I know:

1) Dehn surgery on framed link in $S^3$.

2) Homeomorphism of the surface generate 3-manifold by gluing solid handlebodies along this homeomorphism - Heegaard splitting.

3) Gluing faces of polyhedron (is it somehow formalized ?).

4) Define real smooth function $f$ on $\mathbb R^4$, then $f^{-1}(p)$ is 3-manifold for not critical value $p$ (see Morse theory). More general try smooth function $\mathbb R^5 \to \mathbb R^2$.

5) Spherical 3-manifolds can be determined by finite subgroup of $SO(4)$.

6) Brieskorn manifolds looks as a nice way to define 3-manifold. Is there any generalization to obtain 3-manifolds by intersecting complex manifold of complex dimension 2 with hyperplane or sphere.

Such methods are preferred which could lead to step-by-step "manufacturing" of any 3-manifold. Methods 1) and 2) above satisfy this criteria. Method 1) is the best since we can visualize the link easily in dimension 3 and even 2 as link diagram with numbers.

Note: I have not mentioned hyperbolic manifolds, since I know very little about it.

• There was a good talk by Gromov called "What is a manifold" (see it here: youtube.com/watch?v=u5DLpAqX4YA) where he discusses various ways we know how to get manifolds including some of the above. I think you have a few more that are specific to dimension 3, but his point is interesting in that there are a lot of 'generic' manifolds we really don't have good ways to construct and understand. Of course, with geometrisation, and virtual Haken etc conjectures settled, we have a much better grip for 3-manifolds than in general. – theHigherGeometer Mar 9 '16 at 10:43
• I also edited your title a bit for English grammar, and added the diff-geom tag -- I hope you don't mind. – theHigherGeometer Mar 9 '16 at 10:46
• Should this be this big-list? communiti-wiki? Since I'm here, let me mentioned branched covers, splicing, and plumbings. – Marco Golla Mar 9 '16 at 12:04
• One can also represent any 3-manifold as: a triangulation, an open book, a spine, a planaer linkage, a handle body. – Marc Kegel Mar 9 '16 at 15:10
• There is a variation of the Heegaard splitting (2) that is not as useful, but can help in some situations: Glue up a single handlebody using a fixed-point free involution on the boundary, – Douglas Zare Mar 9 '16 at 18:48

This question ought to be community-wiki, since there is no correct answer.

As indicated in the comments, one may represent a 3-manifold as a triangulation (simplicial complex), special spine, branched cover, real algebraic variety, mechanical linkage,.... In fact, Thurston showed the existence of universal links", such that any 3-manifold is a branched cover over $S^3$ with branch locus the link (such as the figure 8 knot). It is known that any closed 3-manifold is a 3-fold branched cover over $S^3$. Also, the Borromean rings with orbifold locus of order 4 along each component were shown to be a universal orbifold.

There are various combinatorial encodings of triangulations called gems and crystallizations.

Turaev introduced a notion of shadows to describe 3-manifolds. For relations with other presentations of 3-manifolds, see the paper of Costantino and Thurston.

A quite powerful way related to 5) to represent closed 3-manifolds is via the geometrization theorem. One considers orientable 3-manifolds (with torus boundary) admitting geometric structures (on the interiors), and then glue together by homemorphisms of tori (or to Klein bottles), and then connect sums. Although it is somewhat trickier to describe geometric 3-manifolds this way, the advantage is that one may reproduce 3-manifolds without duplication (if done with care), unlike the other techniques.

2) is called a Heegaard splitting.

For 3), there are notions of polyhedral representations of 3-manifolds (even allowing self-gluing of faces). For example, Cannon, Floyd and Parry showed that any 3-manifold admits a "bi-twist" description.

Not every 3-manifold may be obtained by 4).

6) is an interesting question: links of singularities are restrictive, but I'm not sure what's known about intersecting with a general hyperplane or sphere.

• Thank you very much for the answer and for comments to my question. I don't know many of the terms used such as splicing or plumbing. I can improve 4) by considering smooth function $\mathbb R^5 \to \mathbb R^2$. I am keen on extending 6) to quaternion polynomials. – user21230 Mar 11 '16 at 8:07
• Regarding 3), I looked into "bi-twist" paper. I understood that using Heegaard splitting we can obtain any 3-manifold by "face pairing" process on some polyhedron. – user21230 Mar 11 '16 at 8:21
• @MarekMitros: that sounds like an interesting question about quaternionic polynomials. So you have a quaternion polynomial in 2 variables, whose zero set is 4-dimensional, with an isolated singularity, and you want to understand what 3-manifolds can occur as the link of the singularity? – Ian Agol Mar 11 '16 at 13:50
• Yes, this is my plan number 1. The plan number 2 is to look at submanifold in $\mathbb HP^2$ defined by zeros of polynomials of shape $x \overline y + y \overline z$. I don't know how to define real hyperplane on $\mathbb HP^2$ in good way. To make any progress I need first to understand the homology spheres obtained as Brieskorn manifolds. – user21230 Mar 11 '16 at 15:40
• I propose following definition of "real hyperplane" in $\mathbb CP^3 or \mathbb HP^2$. Say for $\mathbb CP^3$ there will be following definition: $K=\{[r(1,v)]: v \in \mathbb C^3, r \in \mathbb R, r^2=1/2\}$ – user21230 Mar 14 '16 at 10:36