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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.

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    $\begingroup$ MO292525 is a recent related question about topological 4-manifolds. Just putting it here to highlight that such questions are still quite non-trivial. $\endgroup$ – Igor Khavkine Feb 21 '18 at 9:29
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    $\begingroup$ If you want to know what information is needed to classify topological 4-manifolds (with some restrictions) the book by Freedman and Quinn is a good start. Wikipedia's 4-manifolds page also leads you down a good initial path. $\endgroup$ – Ryan Budney Feb 21 '18 at 18:21
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    $\begingroup$ @RyanBudney Thank you for the reference, does this book contain the developments explained in the answers below? $\endgroup$ – QGravity Feb 21 '18 at 21:10
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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.

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    $\begingroup$ Regarding your last sentence, for all manifolds of dimension $\geq 3$, $\pi_1(M) = \pi_1(M- pt)$. In addition, open 4-manifolds can have an arbitrary countable fundamental group. $\endgroup$ – Daniele Zuddas Feb 21 '18 at 10:13
  • $\begingroup$ So, isn't it possible to classify finitely-presented groups just like finitely-generated Fuchsian groups? $\endgroup$ – QGravity Feb 21 '18 at 21:01
  • $\begingroup$ In physics, the people are more interested to consider compact four-manifolds with boundaries (which might have some sort of "topological singularity" representing singular behavior in space-time like black hole singularity or big bang). I was very curious about the development in the topological classification of such four-manifolds. $\endgroup$ – QGravity Feb 21 '18 at 21:08
  • $\begingroup$ @QGravity en.wikipedia.org/wiki/… $\endgroup$ – Paul Siegel Feb 21 '18 at 21:56
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    $\begingroup$ @QGravity If a 4-manifold is globally hyperbolic, then it is diffeomorphic to $M\times\mathbb R$ where $M$ is a 3-manifold (whose classification is more tractible), see en.wikipedia.org/wiki/Globally_hyperbolic_manifold. $\endgroup$ – Fan Zheng Jul 22 '18 at 4:09
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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.

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  • $\begingroup$ So, isn't it possible to classify finitely-presented groups just like finitely-generated Fuchsian groups? $\endgroup$ – QGravity Feb 21 '18 at 21:01
  • $\begingroup$ In physics, the people are more interested to consider compact four-manifolds with boundaries (which might have some sort of "topological singularity" representing singular behavior in space-time like black hole singularity or big bang). I was very curious about the development in the topological classification of such four-manifolds. $\endgroup$ – QGravity Feb 21 '18 at 21:07
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    $\begingroup$ It is a classical result that finitely presented groups cannot be algorithmically classified. Then, it is not possible to classify smooth 4-manifolds if you do not put some (actually strong) topological or geometric restrictions. The same holds for 4-manifolds with boundary. $\endgroup$ – Daniele Zuddas Feb 22 '18 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 four-manifolds regarded as space-time in physics. I only heard that the only constraint is that the space-time can be written as $\mathbb{R}\times X$, where $X$ is a compact three-manifold and consider the topological classification of compact three-manifolds (possibly with boundary), but I am not sure. $\endgroup$ – QGravity Feb 25 '18 at 0:46

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