Maybe people understood better why combinatorial approaches don’t prove the Poincare conjecture during the century that the Poincare conjecture was open. :-) I’m not sure that I would agree that it’s hopeless to make the Ricci flow methods combinatorial. At the moment it could seem unlikely or wrong-minded, but hopelessness could be too strong. For instance, circle-packing theorems are a combinatorial analogue of uniformization of surfaces, and there is even a combinatorial analogue of Ricci flow for imperfect circle packings. Minimal surface theory in 3-manifolds also has a combinatorial analogue called normal surface theory. Some of these combinatorial analogues are elegant or important in their own right. Others somehow seem to be miss the point, and just adulterate the geometry.

There were several stages at which manifold topology, or 3-manifold topology, was revolutionized and people thought that a proof of the Poincare conjecture might come soon. One instance was when Papakyriakopoulos proved the loop and sphere theorems. The sphere theorem is lurking behind your discussion of pi_2: The theorem says that pi_2 is generated by embedded spheres. Another was when Smale proved (a version of) the Poincare conjecture in high dimensions. Another was when Thurston introduced geometrization and geometrized a large class of manifolds.

You could say that all of the important combinatorial arguments in 3-manifold topology depend strongly on the use of surfaces. (Or in some cases, foliations or similar.) This smacks of truism, but maybe there is some depth to it. After all, it is vastly easier to prove the analogue of the Poincare conjecture for a 3-torus; it was done by Waldhausen. You do it by cutting the putative 3-torus along what is known as an essential surface. You can repeat that until the manifold becomes a cube with opposite sides glued. Papakyriakopoulos’ theorems are the underlying engine of the argument. Of course you can now also use Ricci flow, which does not clearly rely on Papakyriakopoulos.

No one other than Hamilton and Perelman have thought of a way to analyze the topology of a manifold that does not have nice surfaces in it. On this page, you call the map from S^3 or the swept out form “immersed”, but it actually isn’t immersed; it can have nasty non-immersion singularities. These singularities largely defy combinatorial understanding, but that may not be true forever.

Thurston also used essential surfaces in the part of geometrization that he established. Meanwhile Smale’s work led to surgery theory. It eventually became clear that 5 and more dimensions are very different from 4 and fewer. Topological 4-manifolds behave like high-dimensional manifolds in some ways, but 3-manifolds and smooth (equivalently, PL) 4-manifolds are quite different.

]]>That’s a very nice proof of Lemma 1, which avoids homology theory almost completely! (Though I admit that I was only able to verify that fg had degree 1 by using the fundamental class as in my proof of the lemma.)

It is of course very tempting to try to use the combinatorial description of a 3-manifold as a polytope with various faces identified to attack the Poincare conjecture; there of course have been many failed attempts to do so, and I wonder if there is now some sort of understanding as to why this is the case. (Certainly it is clear that it is hopeless to try to transplant Ricci flow methods to the combinatorial setting.)

]]>First, yes, if you want to state lemma 1 in the generality of topological manifolds, you might as well start with the triangulation theorem. The obvious generalization of the result to n dimensions does not require a triangulation (even though, in the end, the Poincare conjecture is true and so there is a triangulation), but the rest of the argument is then less intuitive and requires more abstract appeals to Hurewicz, Whitehead, and Flexner.

Anyway, start with a triangulation and consider the more general method of building a manifold by gluing together polytopes face to face. Then you can always reduce the number n-cells to 1 by knocking out walls. In 3 dimensions, then, every 3-manifold is obtained by gluing a single polyhedron to itself.

A slightly stronger version of Lemma 1 as stated here is that if pi_1(M) and pi_2(M) both vanish, then M is homotopy equivalent to S^3. Since pi_1 and pi_2 both vanish, you can homotop the identity map on M to collapse its 2-skeleton to a point. This collapse of M is plainly homeomorphic to S^3. So you immediately get pair of maps f:M -> S^3 and g:S^3 -> M such that the composition gf is homotopic to the identity. The other composition is also homotopic to the identity because it has degree 1. Finis.

With a little more hands-on work, and with one non-trivial application of the Hurewicz theorem, you can relax lemma 1 to show that if pi_1(M) vanishes, then pi_2(M) also vanishes. The cellulation of M can be simplified a bit further: You can identify all of the vertices by contracting edges. If you then compare the cellular presentations of pi_1 and H_1, you immediately get the k=1 case of the Hurewicz theorem, because the presentations are identical except that H_1 is abelian. You also obtain Poincare duality between H_1 and H_2 as follows. The Euler formula tells you that there are the same number of 1-cells and 2-cells. The boundary map from 2-cells to 1-cells is some matrix M, and the cellular homology computation says that H_2 is the kernel of M while H_1 is the cokernel. (Since all 1-cells are loops and all 2-cells are two sides of the same 3-cell, the other two boundary maps vanish.) So obviously H_2 is free and has the same rank as the free part of H_1. Finally the Hurewicz theorem says that if pi_1 vanishes, then pi_2 = H_2.

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Finally here are some remarks about how to prove either Flexner’s theorem or triangulate 3-manifolds. The basic lesson is that topological manifolds are a hard start. Flexner’s theorem proceeds by embedding the manifold M in a high-dimensional sphere or Euclidean space. It is then an intersection of polyhedral regions, and you can show (by Alexander duality) that its Cech cohomology is suitably dual to the homology of its complement. Then, separately, since M is a manifold, its Cech cohomology is isomorphic to its singular cohomology. You can also relate the homology of M to the homology of its complement using gluing axioms of homology theory. This sketch does not really do the argument justice, but it is the way to get started.

Or, if you want to triangulate 3-manifolds, there is a modern proof due to Andrew Hamilton based on the Kirby-Siebenmann method to triangulate high-dimensional manifolds (when that is possible). The proof is fairly short and fairly concrete, but it’s still quite tricky — the Kirby-Siebenmann work in general requires a number of exacting inferences. The proof also depends on a non-trivial 3-manifold classification result due to Waldhausen: if a PL 3-manifold is irreducible and homotopy equivalent to a 3-torus, then it is a PL 3-torus. It comes full circle, because Perelman reproves this!

]]>So you first need to use the relative Hurewicz theorem to conclude that your map is indeed an isomorphism on homotopy groups. Here the fact that M and S^3 are simply connected is an important condition; as you mention, the PoincarĂ© homology sphere P is not simply connected, and yet the construction you describe gives a map which is a homology isomorphism.

To be precise, a map between simply-connected spaces which is an isomorphism on integral homology groups is an isomorphism on homotopy groups, by the relative Hurewicz theorem; and a map between connected CW complexes which is an isomorphism on homotopy groups is a homotopy equivalence, by Whitehead’s theorem.

]]>Since the bulk of what you are doing has to be smooth anyway, I would say that it’s incongruous to invoke Moise’s theorem more than at the very beginning, because it is relatively easy to find a smooth triangulation of a smooth manifold. It’s even easier to make a fundamental class if that is all you want: Take the cellulation induced by a Morse function and take the sum of the top-dimensional cells. But okay, the point has been made.

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While I am at it, I’d like to air an open problem in geometric analysis that is related to existence of smooth structures. Allan Hatcher showed that a PL 4-manifold has a unique smooth structure. The key technical result is that the diffeomorphism group of the 3-sphere is homotopy equivalent to the isometry group O(4). Or, it turns out to be enough to show that the space of smooth fibrations of a cylinder S^2 x I by spheres is contractible. Hatcher’s proof of this (I think) largely combinatorial in spirit, and also some people have said that the proof is quite complicated. It would be interesting to prove it using geometric analysis.

A natural idea that comes to mind is one that actually works in the previous dimension: Use mean curvature flow to flatten all of the leaves of the fibration simultaneously. The flow cannot make the leaves cross. However, in 3 dimensions, mean curvature flow can develop singularities in a finite amount of time, so this construction doesn’t work. I have been wondering whether you can replace mean curvature flow by another parabolic flow that (optimistically) can’t develop singularities quickly enough to wreck the argument. Mean curvature flow is the gradient flow of th area functional. What if instead you took the gradient flow of the integral of exp(R), where R is the scalar curvature of the surface? At least the simplest sort of singularity, a shrinking hyperboloid, might not be able to form in a finite amount of time. In this context, it would approximately pinch off a bulb from the sphere, and the bulb would (if we’re lucky enough) shrink away first.

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