4-Manifolds of which there exist no Kirby diagrams

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.

Solutions Collecting From Web of "4-Manifolds of which there exist no Kirby diagrams"

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.