4
$\begingroup$

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.

  • 0
    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 decomposition2011-07-27

1 Answers 1

4

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.