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.

Answer 2

As Paul Siegel pointed out, the fundamental group of a smooth closed orientable 4-manifold can be an arbitrary finitely presented group, and for this reason a general classification is not possible, unless, possibly, for some classes of fundamental groups, e.g. the trivial group, by Freedman's work. However, the known classifications, even with this restriction, are in the topological category, while there is no complete classification, even for a fixed topological type, in the smooth (or PL) category. For example, nothing is known about the smooth classification of smooth 4-manifolds homeomorphic to the 4-sphere (the smooth 4-dimensional Poincaré conjecture says that there is just one, namely the standard $S^4$).

Regarding your last question, in the smooth category there are handle decompositions, which allow to build compact smooth 4-manifolds as the union of finitely many $k$-handles, $0\leq k\leq 4$, where a 4-dimensional $k$-handle ($k$ is the index of the handle) is a copy of $B^k \times B^{4-k}$ attached along $S^{k-1} \times B^{4-k}$ to the boundary of a given 4-manifold. If the manifold is closed and connected, then there is such a handle decomposition with only one 0-handle and one 4-handle. A theorem of Poenaru and Laudenbach helps in making this presentation effective: any smooth closed connected oriented 4-manifold can be reconstructed, uniquely up to diffeomorphisms, from a handle decomposition where only the handles up to index 2 are given (in other words, you do not need to know 3- and 4-handles to determine the closed manifold). There is also a nonorientable version of this.

This information can be encoded in a Kirby diagram, which provides a finite presentation of any smooth closed 4-manifolds. In the TOP category more work is needed, see the MO question mentioned by Igor.