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I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be triangulated, using the technique of the Seiberg-Witten monopole equation.

As a physicist I'm quite puzzled how the monopole equation, which seems to require (at least to me) a smooth structure, can say something about topological manifolds that can't be triangulated.

I would appreciate if some experts could guide me through this confusion of mine.

Let me say a bit more about my confusion.

In a classic situation of Donaldson proving that some 4d topological manifolds don't have smooth structure, he showed using gauge theory that the intersection form of smooth manifolds are such and such, and then it's clear if there is a topological manifold whose intersection form isn't one of such and such, it doesn't have any smooth structure.

So I think I can understand that the gauge theory can be used to answer the (non)existence of smooth structure... But how does it apply to whether a manifold can be triangulated?

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For $4$-manifolds, smooth = $PL$. That's an old result not related to gauge theory. – Alex Degtyarev Mar 23 '14 at 9:14
up vote 25 down vote accepted

The very short answer is that there is no direct connection between gauge theory (which is living on some perhaps hypothetical smooth 4-manifold) and triangulation of some high-dimensional topological manifold. Here are some remarks to justify that statement.

The question addressed in Quinn's expository article is not whether 4-manifolds can be triangulated, or even if they are PL. The question is whether every manifold in any dimension (for this answer, manifold means merely topological manifold) is homeomorphic to a simplicial complex. The issue of whether manifolds are PL (meaning that there is a triangulation in which the link of every simplex is a PL sphere) was settled by Kirby and Siebenmann in the late 1970's, building on the work of many other people; in particular in all dimensions higher than four there are non-PL manifolds. The existence of non-PL (equivalently, in this dimension, non-smooth) 4-manifolds was shown later by Freedman.

The difference between PL and triangulable was highlighted in a prescient article of Siebenmann (Are non-triangulable manifolds triangulable?, available as an appendix to the book of Kirby and Siebenmann). Besides simply drawing attention to the problem, he pointed out the relevance of the double suspension conjecture (now a theorem) and Rohlin's theorem. Some years later, independent work of Galewski-Stern (see Quinn's bibliography) and T.Matumoto (surprisingly, not cited by Quinn, but see Triangulation of manifolds, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pp. 3–6, 1978.) reduced the existence question for manifolds of any dimension to a question about the homology cobordism group of homology 3-spheres. The specific issue was whether there is a homology $3$-sphere $Y \# Y$ with nontrivial Rohlin invariant such that $Y \# Y$ is the boundary of an acyclic $4$-manifold. This was recently resolved by Manolescu ( who showed that there is no such $Y$. I recommend the introduction to Manolescu's paper for some of the background and a discussion of the sophisticated gauge-theoretic techniques that he uses.

The question of `why does it work' is really complicated; the shortest summary I can make is that Galewski-Stern/Matumoto set up an obstruction theory analogous to that found in smoothing theory or the work of Kirby-Siebenmann. As in the older result, this obstruction theory is built on foundational geometric techniques whose development span the 1960's and 1970's. As in the work of Kirby-Siebenmann, the upshot of this is that there's an essential unique obstruction that (after lots of technical work) comes down to Rohlin's theorem on the signature of a spin 4-manifold and the aforementioned problem about $4$-dimensional homology cobordisms. Manolescu used refined arguments in gauge theory to solve this problem.

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