Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin structure, etc.).

This raises a couple of questiona:

1.Is any compact or non-compact 4-manifold obtainable as a (finite or infinite) handle diagram ?

2.What are the properties needed for a compact or non-compact 4-manifold to be represented as a handle diagram ?

3.What are examples of 4-manifolds with no handle diagram ?

The diagrams can be as complicated as you want (so 0-, 2-, 3-, 4-) handles can be present. I do not know if you can get rid of all the 3-handles in the non-compact case.

This question came forth from the discussion explicit "exotic" charts . I am trying to get help of more people on that, then putting those things in comments (the question of explicit charts of an $\mathbb{E}\mathbb{R}^4$ is another one, albeit interesting in it's own right).

The question is answered by Bob Gompf by email, see my comment for the main part of his answer.

share|cite|improve this question
A related question on math overflow:… – Grumpy Parsnip Jul 25 '11 at 16:06
Usually one finds a handle decomposition via a Morse function, but these don't behave so well in the noncompact case, so something subtle is going on. A quick Google search turned up this reference… which states that there are Morse functions on every noncompact manifold with no critical points.! – Grumpy Parsnip Jul 25 '11 at 16:24
I have received a mail from Bob Gompf on this, seems that indeed every 4-manifold has a Kirby diagram, so my question is answered by external source. – Willem Noorduin Jul 27 '11 at 12:31
The problem is that one cannot draw such a thing easily (but is does exist). There is also the problem of attaching infinitely many handles to its 0-handle, which can be fixed either by adding a collar to the boundary along with each handle, or introduceing canceling handle pairs so that there are infinitely many 0-handles. – Willem Noorduin Jul 27 '11 at 12:41
In particular, Bob doesm't know any handle diagrams of large exotic R^4's. All known examples of these require infinitely many 3-handles in their handle decomposition – Willem Noorduin Jul 27 '11 at 12:47

Weird to answer your own question, but one becomes wiser with years. Seens that every 4-manifold can be represented as a Kirby Diagram. Problem is that these things can get very complicated (infinite many 1- or 3-handles, or infinite 0-handles, kinks in the handles, etc). So the question can be answered negatively: there are none.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.