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 fourmanifolds possibly with boundaries or even open fourmanifolds. More concretely, I am wondering to which extent the work of Michael Freedman classifies the topological fourmanifolds, and what information is required to uniquely specify the topological class of a compact/noncompact fourmanifold.

3$\begingroup$ MO292525 is a recent related question about topological 4manifolds. Just putting it here to highlight that such questions are still quite nontrivial. $\endgroup$– Igor KhavkineFeb 21, 2018 at 9:29

3$\begingroup$ If you want to know what information is needed to classify topological 4manifolds (with some restrictions) the book by Freedman and Quinn is a good start. Wikipedia's 4manifolds page also leads you down a good initial path. $\endgroup$– Ryan BudneyFeb 21, 2018 at 18:21

1$\begingroup$ @RyanBudney Thank you for the reference, does this book contain the developments explained in the answers below? $\endgroup$– QGravityFeb 21, 2018 at 21:10

1$\begingroup$ Pointing out a paper that may be relevant (even to show that this question is welldefined). arxiv.org/abs/1810.05988 $\endgroup$– Ian AgolNov 6, 2023 at 19:36
2 Answers
Suppose you can classify all open 4manifolds. In particular you can classify all manifolds of the form $M^4  pt$ where $M^4$ is a closed 4manifold, and consequently you can classify all closed 4manifolds. But this classification problem reduces to the word problem on a finitely presented group (every such group is the fundamental group of a closed 4manifold) and this is known to have no algorithmic solution.
Freedman's work solves the classification problem for closed simply connected 4manifolds  it says that the intersection form on degree 2 homology together with the KirbySiebenmann class provide a complete invariant for such manifolds. Freedman's techniques can also be used to produce complete invariants for closed 4manifolds 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 4manifolds, 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.

6$\begingroup$ Regarding your last sentence, for all manifolds of dimension $\geq 3$, $\pi_1(M) = \pi_1(M pt)$. In addition, open 4manifolds can have an arbitrary countable fundamental group. $\endgroup$ Feb 21, 2018 at 10:13

$\begingroup$ So, isn't it possible to classify finitelypresented groups just like finitelygenerated Fuchsian groups? $\endgroup$– QGravityFeb 21, 2018 at 21:01

$\begingroup$ In physics, the people are more interested to consider compact fourmanifolds with boundaries (which might have some sort of "topological singularity" representing singular behavior in spacetime like black hole singularity or big bang). I was very curious about the development in the topological classification of such fourmanifolds. $\endgroup$– QGravityFeb 21, 2018 at 21:08


1$\begingroup$ @QGravity If a 4manifold is globally hyperbolic, then it is diffeomorphic to $M\times\mathbb R$ where $M$ is a 3manifold (whose classification is more tractible), see en.wikipedia.org/wiki/Globally_hyperbolic_manifold. $\endgroup$ Jul 22, 2018 at 4:09
As Paul Siegel pointed out, the fundamental group of a smooth closed orientable 4manifold 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 4manifolds homeomorphic to the 4sphere (the smooth 4dimensional 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 4manifolds as the union of finitely many $k$handles, $0\leq k\leq 4$, where a 4dimensional $k$handle ($k$ is the index of the handle) is a copy of $B^k \times B^{4k}$ attached along $S^{k1} \times B^{4k}$ to the boundary of a given 4manifold. If the manifold is closed and connected, then there is such a handle decomposition with only one 0handle and one 4handle. A theorem of Poenaru and Laudenbach helps in making this presentation effective: any smooth closed connected oriented 4manifold 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 4handles 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 4manifolds. In the TOP category more work is needed, see the MO question mentioned by Igor.

$\begingroup$ So, isn't it possible to classify finitelypresented groups just like finitelygenerated Fuchsian groups? $\endgroup$– QGravityFeb 21, 2018 at 21:01

$\begingroup$ In physics, the people are more interested to consider compact fourmanifolds with boundaries (which might have some sort of "topological singularity" representing singular behavior in spacetime like black hole singularity or big bang). I was very curious about the development in the topological classification of such fourmanifolds. $\endgroup$– QGravityFeb 21, 2018 at 21:07

1$\begingroup$ It is a classical result that finitely presented groups cannot be algorithmically classified. Then, it is not possible to classify smooth 4manifolds if you do not put some (actually strong) topological or geometric restrictions. The same holds for 4manifolds with boundary. $\endgroup$ Feb 22, 2018 at 9:57

$\begingroup$ Thank you very much, your answer and comment are very helpful. Based on your comment, I think one then need to understand what sort of topological constraints come from the global hyperbolicity condition on fourmanifolds regarded as spacetime in physics. I only heard that the only constraint is that the spacetime can be written as $\mathbb{R}\times X$, where $X$ is a compact threemanifold and consider the topological classification of compact threemanifolds (possibly with boundary), but I am not sure. $\endgroup$– QGravityFeb 25, 2018 at 0:46