TITLE: Topological Classification of Four-Manifolds
QUESTION [7 upvotes]: 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.

REPLY [6 votes]: 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.