[Math] Topological Classification of Four-Manifolds

Question

It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. I was wondering about the similar classification for a general compact four-manifolds possibly with boundaries or even open four-manifolds. More concretely, I am wondering to which extent the work of Michael Freedman classifies the topological four-manifolds, and what information is required to uniquely specify the topological class of a compact/non-compact four-manifold.

Answer 1

Suppose you can classify all open 4-manifolds. In particular you can classify all manifolds of the form $M^4 - pt$ where $M^4$ is a closed 4-manifold, and consequently you can classify all closed 4-manifolds. But this classification problem reduces to the word problem on a finitely presented group (every such group is the fundamental group of a closed 4-manifold) and this is known to have no algorithmic solution.

Freedman's work solves the classification problem for closed simply connected 4-manifolds - it says that the intersection form on degree 2 homology together with the Kirby-Siebenmann class provide a complete invariant for such manifolds. Freedman's techniques can also be used to produce complete invariants for closed 4-manifolds with certain prescribed fundamental groups, but this of course depends on the group. So I can't disprove the possibility that there is a classification of simply connected 4-manifolds, but on the other hand I think the fundamental group of a closed $M^4$ with simply connected $M^4 - pt$ can still be quite complicated, so I'm not sure.