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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.

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A related question on math overflow: mathoverflow.net/questions/54143/… –  Grumpy Parsnip Jul 25 '11 at 16:06
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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 books.google.com/… 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
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1 Answer

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.

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