Finite data for 4manifolds: I have a somewhat hazy memory of seeing a Ph.D. thesis around 2005 which showed that any compact topological 4manifold $M$ can be specified by a finite amount of data. The very rough idea is to exploit the fact that $M \setminus pt$ ($M$ minus a point) can be triangulated and explicitly give such a triangulation for a large initial segment $X$ of $M \setminus pt$, and an even larger initial segment $Y$ of $M \setminus pt$, together with a certificate that $Y \setminus X$ contains a topologically flat embedded S^3 cutting off the end of $M \setminus pt$. This data allows you to build the closed topological manifold $M$ and argue that the result is unique. The important detail is how to give the required certificate. Over the years several people have asked me about this “finite data for 4manifold” question. I would like to locate the reference, or failing that find time to write up a proof.

$\begingroup$ Mpt means $M$ minus a point? $\endgroup$ – YCor Feb 8 '18 at 23:24

$\begingroup$ Yes. I've edited the question to make that clearer. $\endgroup$ – Kevin Walker Feb 8 '18 at 23:58

$\begingroup$ One approach is to use the alphaapproximation theorem. What's unclear however is how effective are the estimates. $\endgroup$ – Misha Feb 9 '18 at 21:50

$\begingroup$ Can any closed topological 4manifold be smoothed by removing a fake 4ball? Do homology spheres bound unique fake 4balls up to homeomorphism? $\endgroup$ – Ian Agol Feb 11 '18 at 14:37
I understand what you're getting at, but I think the statement "any compact topological 4manifold can be specified by a finite amount of data" has a trivial answer (modulo the literature). Cheeger and Kister proved that compact 4manifolds are countable. So given a bijection of 4manifolds with $\mathbb{N}$, each 4manifold can be specified by an integer.
Of course, this is a cheeky answer, and not what your asking for. The content of your question is asking for a specific way to specify a 4manifold, which is tied up in the certificate for a 3sphere. I'm not aware of this work. But maybe the proof of CheegerKister's theorem could be turned into a way to specify a 4manifold?
Here's a possible example of how such a specification might go. A compact manifold can be specified by a finite collection of closed balls and embeddings between them, so that the equivalence class generated by these embeddings with the quotient topology gives the manifold. Embeddings of balls may be approximated by PL maps (which might no longer be embeddings, but should have the property that the set of double points is as small diameter as one please). Maybe there's some sort of criterion which would say that a collection of such maps between balls uniquely approximates a set of embeddings which gives a manifold? An existence criterion of this sort sounds trickier to me than a uniqueness criterion.

2$\begingroup$ For me "specify" should mean one can check that the data presented in fact defines a manifold, and also, from that data there should be an algorithm that with increasing accuracy defines the points of a representative manifold in the homeomorphism type. The idea of using the CheegerKister theorem, based on the EdwardsKirby local contractibility of spaces of homeomorphisms, is quite interesting but the technical issue look daunting, at least to me. $\endgroup$ – Michael Freedman Feb 9 '18 at 19:55

$\begingroup$ ...compact 4manifolds are countable. Homeomorphism classes of compact 4manifolds? Otherwise we'd have some insight into the smooth Poincaré conjecture in dimension 4. $\endgroup$ – David Roberts Feb 10 '18 at 2:42

1$\begingroup$ Yes, homeomorphism classes, I should have clarified. For smooth manifolds, countability is even easier, since they are triangulable. But there are uncountably many smooth $R^4$s, and uncountably many homeomorphism classes of noncompact manifolds. $\endgroup$ – Ian Agol Feb 10 '18 at 3:44
I happened to discuss this very same question with Fico last December. Maybe he is the one who was asking you.
What's written below makes sense in any dimension, but dimension 4 is the most interesting since one cannot use the results of Kirby and Siebenmann on existence of a handle decomposition. Note that a handle decomposition does not yet give a finitely description of a topological manifold since ones to specify the attaching maps. The same idea of a $\delta$manifold allows one to deal with this issue by replacing homeomorphic attaching maps with $\delta$homotopy equivalences of suitable subcomplexes.
Here is an approach using the $\alpha$approximation theorem in dimension 4 (Theorem 4 in "Curvature, tangentiality, and controlled topology" of Ferry and Weinberger, Inventions, 1991): if $M, N$ are compact 4manifolds with boundary and $f: M\to N$ is $\delta$homotopyequivalence (with $\delta$ sufficiently small) which restricts to a homeomorphism on the boundary, then $f$ is $\epsilon$homotopic to a homeomorphism. Here I am assuming that $M$ and $N$ are metrized; a $\delta$homotopyeqquivalence is a proper homotopyequivalence such that tracks of homotopies to the identity have diameter $\le \delta$.
What's unclear to me is how effective is $\delta$ in this theorem, provided that, say, $M, N$ are triangulated and $f\partial M$ is a PL homeomorphism.
Remark. I do not know if $B_{ij}$'s can be taken to be topological balls, this is related by my question about existence of good covers of topological manifolds.
Setting this issue aside, here is how one can use this theorem. A compact topological 4manifold can be defined in terms of a finite collection of 4balls $B_i$ and partially defined gluing homeomorphisms $$ f_{ij}: B_{ij}\subset B_i\to B_{ji}\subset B_j. $$ I will replace these with a combinatorial data: triangulated 4balls $C_j$ are piecewiselinear maps $g_{ij}: C_{ij}\to C_{ji}$. Here I am assuming that the subcomplexes $C_{ij}, C_{ji}$ are equipped with refinements of the original triangulations coming from $C_i, C_j$, so that each $g_{ij}$ is linear on each simplex.
The triangulations of $C_i$'s yield piecewiseEuclidean metrics on these balls in terms of which I will define lengths of tracks of homotopies (which I assume to be piecewiselinear).
Let $X$ be the complex obtained by gluing $C_i$'s via the $g_{ij}$'s.
Definition. $X$ is a $\delta$manifold if each $g_{ij}$ restricts to a homeomorphism $$ \partial C_{ij}\to \partial C_{ji} $$ and $g_{ij}$ is a $\delta$homotopyequivalence $C_{ij}\to C_{ji}$.
The point of this definition is that (compact) each $\delta$manifold is a combinatorial creature and is encoded via a finite amount of data.
The alphaapproximation theorem then says that if $\delta$ is small enough (depending on what the $C_i, C_{ij}$'s are) then each $\delta$manifold is homotopyequivalent to a compact 4dimensional topological manifold $M_X$ and, moreover, this manifold is unique up to a homeomorphism.
Furthermore, given a compact topological 4manifold, represented via a system of gluing homeomorphisms $f_{ij}: B_{ij}\to B_{ji}$, one approximates these homeomorphisms via PL maps which are $\delta$homotopyequivalences for arbitrarily small $\delta$. (There are some, possibly nasty, technicalities hidden here since the original maps $f_{ij}$ are defined on open subsets and those have to be replaced by slightly smaller finite subcomplexes.) Hence, every compact topological 4manifold $M$ yields a $\delta$manifold $X$.


$\begingroup$ @Ian Agol: This is unclear. It is related to my MO question about existence of good covers for topological 4manifolds, which appears to be an open problem. $\endgroup$ – Misha Feb 10 '18 at 8:59

$\begingroup$ In a way what I write is similar to the proposal in the post: One can cover $M$ by two complexes instead of several. This might provide a useful simplification. $\endgroup$ – Misha Feb 10 '18 at 9:28
In my understanding, I guess that the following strategy could be be attempted. Let $M$ be a closed connected orientable 4manifold (while nonorientable 4manifolds are doubly covered by orientable ones). Surgerying $M$ along finitely many embedded tame circles produces a simply connected 4manifold $M'$. Then, $M$ can be recovered from a certain simply connected manifold by surgerying finitely many pairwise disjoint locally flat 2spheres with trivial normal bundle. Now, simply connected 4manifolds are classified by Mike Freedman's celebrated theorem in terms of the intersection form and the KirbySiebenmann invariant (which amounts to finite data). It remains to understand how much data we need to determine the embedded spheres to be surgered in a simply connected 4manifold. Of course, we need more than homotopy (= homology since $\pi_1 = 0$) classes.

1$\begingroup$ This is quite different from the argument I sketch but I like it. Indeed the important detail is giving a finite description of the imbedded 2spheres building on a finite description of the 1connected manifold. This is a little tricky, and as far as I see, will still need some work. ( I also do not know how to make the deltamanifold idea precise.) I greatly appreciate the input from Ian, Misha, and Daniele. If the reference I was looking for does not turn up in a couple weeks I will write up something. My present understanding is not succinct enough to just write it out on Mathoverflow. $\endgroup$ – Michael Freedman Feb 12 '18 at 0:53