This past week, Ciprian Manolescu posted a preprint on ArXiv proving (allegedly- I haven’t read the paper beyond the introduction) that the Triangulation Conjecture is false.
This is big news. I feel it’s the last nail in the coffin of the Hauptvermutung. I’d like to tell you a little bit about the conjecture, and about Manolescu’s strategy, and what it has to do with low dimensional topology.
What is an -manifold?
“That’s a silly question,” someone might say. “It’s a (bla bla bla) space that is locally homeomorphic to ”.
Ah, but homeomorphic via what kind of map? A continuous map? A smooth map? A piecewise linear map? In two dimensions it makes no difference. But in higher dimensions, the concept of an -manifold turns out to be more subtle than the case might lead you to believe.
For Poincaré in 1895, a `three manifold’ starts out as a quotient of by actions of certain groups with a cube as fundamental region. Later in Analysis Situs, he presents a more general notion of a manifold by identifying faces of polyhedra. Poincaré’s manifolds were supposed to be differentiable, but his constructions of them are PL. It seems that Poincaré considered that there was no difference. Obviously all corners can be smoothed and all smooth structures can be approximated by broken linear structures, and (although this notion came later) obviously topological manifolds can all be triangulated. Obvious, but false. The shocking truth is that smooth, PL, and topological categories diverge sharply, even if we restrict ourselves to thinking about individual manifolds and not maps between manifolds.
The following explanation is distilled from Ranicki’s excellent survey On the Hauptvermutung.
A triangulation of a topological space is a simplicial complex together with a homeomorphism from the polyhedron of to . A space is triangulable if it admits a triangulation. In 1908, Steinitz and (independently) Tietze formulated the Main Conjecture, or Hauptvermutung, which states that triangulations of homeomorphic spaces are combinatorially equivalent i.e. become isomorphic after subdivision. In 1961, Milnor shocked the mathematical world by finding a counterexample. But his counterexamples were not manifolds- and as geometric topologists, we’re only supposed to care about manifolds, right? (I really needed sarcastices for that last sentence). So is the Hauptvermutung at least true for manifolds? Closely related, we have the:
Combinatorial Triangulation Conjecture
Every compact topological manifold can be triangulated by a PL manifold.
In dimension two the conjecture holds by 1925 work of Radó, and in dimension three by work of Moise. But (prepare to be shocked again) it’s false in dimensions by work of Kirby and Seibenmann, and in dimension by work of Freedman.
Poor Hauptvermutung! Can’t it at least be just a little bit right?
Let’s try again… Maybe to be homeomorphic to a PL manifold is too much to ask. Recall that the link of a simplex σ is the complex of all simplices at `distance one’ from σ (i.e. separated by one edge from σ). A PL manifold has the property that the link of each simplex of its triangulation is a sphere. Maybe this is too much to ask. After all, there are plenty of not-terribly-pathological spaces which don’t satisfy this property, such as the double suspension of any non-trivial homology sphere (such as the Poincaré sphere), by Cannon and Edwards’s Double Suspension Theorem. So let’s weaken the Combinatorial Triangulation Conjecture:
Every compact topological manifold can be triangulated by a locally finite simplicial complex.
Unfortunately even this fails in dimension four, with Freedman’s -manifold providing a counterexample… but maybe dimension four is pathological. Maybe it holds in dimension greater than five.
For all the seeming simplicity of the statement, there’s another surprise in store- even though it’s about manifolds in dimension , it actually reduces to a problem in low dimensional topology of smooth three manifolds. It’s a problem about holomogy cobordism of homology spheres, so let’s briefly review that story.
By the way, the paper I first learnt this story from (and which made a deep impression on me) is:
Ruberman, D., & Saveliev, N. (2005). Casson-type invariants in dimension four. Geometry and topology of manifolds, Fields Inst. Commun, 47, 281-306.
Recall that an integral homology sphere is a closed oriented -manifold with . There are plenty of these, the most famous perhaps being the Poincaré homology sphere. A homology cobordism between homology spheres and is a compact oriented -manifold with boundary such that the inclusions induce isomorphisms for . Homology cobordism is an equivalence relation, and the set of equivalence classes with operation the connect sum forms a group .
Next let’s recall the Rokhlin Invariant. Every integral homology sphere is the boundary of a compact spin -manifold , and the Rokhlin invariant is an eighth of the signature of modulo two. It is invariant under homology cobordism, and so defines a homomorphism .
Alright now! Galewski and Stern, and independently Matsumoto, reduced the Triangulation Conjecture to the following:
Equivalent Statement to the Triangulation Conjecture
There exists a homology -sphere with Rokhlin invariant one which is of order two in .
So the goal becomes to understand torsion in (at least for Rokhlin invariant one homology spheres). It turns out not to be easy, probably because it’s a statement about the smooth category, and so many of our techniques are secretly PL. Until the 1980’s, all that was known about was that is an epimorphism. Then, using equivariant gauge theory, Fintushel and Stern showed that has many elements of infinite order, and Furuta showed that it is infinitely generated. But does it have -torsion? How would you even approach a question like that?
Manolescu’s answer is to construct a Seiberg–Witten Floer Homology with all of the symmetries that the Seiberg-Witten equations have in the situation at hand. Namely, because the bounded four manifold has a spin structure, the Seiberg–Witten equations turn out to have a symmetry group known as . Manolescu has experience with equivariant Seiberg–Witten theories, and he constructs a weapon which (to my untrained eye) looks big enough and strong enough for the task. There’s probably a lot more that this Floer homology can do, and one can now expect vigourous progress in its study.
All of this leaves the Hauptvermutung pretty much as dead as dead can be. Topological manifolds in dimension greater than three just can’t be triangulated in general… And we’re going to have to learn to live with it.
Edit: As pointed out in the comments, by Galewski-Stern dimension 5 counterexamples must be non-orientable (which is quite striking!), but in dimension 6 and above, there are orientable counterexamples as well.
Executive summary for the casual mathematical tourist
Given a lego set whose blocks are triangles (i.e. simplices), there are many shapes (i.e. compact topological manifolds) you could build. But could you build all of them?
In dimension 1 you obviously could (a circle can be cut into line segments), in dimension 2 you obviously could (any compact surface can be sliced up into triangles, and you take those to be the blocks). It turns out that you can in dimension 3 as well (intuitively unsurprising, but difficult to prove), but surprisingly you cannot in dimension 4. But we all know that dimension 4, being the dimension in which we live (space plus time) is a bit crazy- what about in dimension 5 and higher? Galewski and Stern showed that you can triangulate any shape (shape= compact topological manifold) in dimension 5 and up (in just about the weakest sense that a decomposition into triangles has any right to call itself a triangulation) if and only if you can triangulate a specific 5-dimensional shape called the Galewski-Stern manifold. Manolescu’s preprint proves you cannot, showing us again that there are more things in heaven and earth, Horacio, than are dreamt of in your (triangulated) philosophy.