Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds really?
This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it really is.
In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually exist, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.
It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s On piecewise linear cell decompositions).
The definition of smooth manifolds has bothered me ever since these excellent slides from a talk by Oleg Viro did their job to make me feel uncomfortable.
The modern definition of differentiable manifolds appeared first in
The Foundations of Differential Geometry by Veblen and Whitehead, Cambridge University Press, 1932. It was inspired by a book on Riemann surfaces by Weyl. It’s an awful definition, really, in that it’s quite long, and it requires a choice of atlas, which plays no role in what follows and is ultimately done away with (in favour of a maximal atlas, perhaps). Worse still, the category of differentiable manifolds is not stable under basic set theoretical operations. This ought to especially bother quantum topologists, because the algebraic categories with which quantum invariants are defined do not reflect this instability. The image of a differentiable manifold under a differentiable map might not be a manifold. The space of maps between manifolds is not a manifold. If you slice up manifolds, as required to construct TQFT invariants, the pieces are not manifolds. Maybe worst of all (depending on your perspective), the most important functor in smooth topology, the tangent functor (which assigns tangent bundle to manifold and map to map so that the obvious diagram commutes) is not a representable functor. I think that I believe Grothendieck’s assertion that useful functors should be representable, so this fact makes me most uncomfortable of all, in a strange sort of way.
The solution is surely to expand the category of manifolds which we consider. So a knot theorist perhaps thinks about tangles instead of knots, or maybe about virtual tangles (or even Dror’s w-knotted objects), and (in the world of manifolds) maybe we think about manifolds with corners, or, more ambitiously, stratified spaces. This lets us chop up manifolds, take quotients by group actions which we care about, contract loops to a point, and take covering spaces branched over graphs; which are all things I care about doing.
But what about spaces of diffeomorphisms between manifolds? Our category of smooth manifolds really ought to have exponential objects, in order to be a Cartesian closed category. I don’t understand enough to know why this would be useful concretely, but on the same fuzzy level with which I’ve been talking up until this point, diffeomorphism groups of manifolds are objects of obvious interest in geometric topology (the mapping class group is the group of connected components of the space of diffeomorphisms of the manifold, for example. Via Heegaard decompositions, a mapping class of a surface gives rise to a -manifold.) It would be nice to be able to discuss diffeomorphism groups on equal footing with smooth manifolds. Instead, they’re typically discussed in the context of tame Fréchet spaces, and the inverse function theorem gets generalized in a non-trivial way; it’s all a bit hard. I don’t know whether a larger category could make such theorems more convenient, or at least conceptually clarify what class of topological spaces we can expect them for, but it would be wonderful if it were so.
When we expand our category, perhaps we are throwing away “reality”… I don’t know. Natural numbers have some sort of metaphysical existence, I believe, and rational numbers sort-of still exist; but real numbers are an abstraction. In what sense can we claim that a number exists that we can never construct?
One interesting attempt to expand the category of smooth spaces to a cartesian closed category is Frölicher spaces. This approach, and the category of diffeologies, which is another good idea which is more general, are extensively discussed by experts at the n-Category Cafe. Philosophically, I don’t know whether or not these smooth spaces “exist” in any Platonic metaphysical sense; but cartesian closedness might be worth a bit of abstraction.
But what about representability of the tangent functor? Here’s where another mesmerizingly beautiful idea comes into play. Synthetic differential geometry. I’d love to talk about it, and I really should (especially if there are genuine links to proper low dimensional topology, which one can exploit); in a nutshell of a nutshell, or “on one leg” as we say in Hebrew, you trade sets for topoi to get a version of smooth topology (and differential geometry) in which infinitesmals in the Liebnitz sense exist, and you have for which . Words like “topos” sound complicated, but actually the core idea meshes well with our naive intuition as humans (although not with our mathematical training)- is it really such a stretch to imagine a circle intersecting its “tangent line” along an infinitesmal line rather than a point? Or to have an infinitesmally small n-manifold for each n?
Synthetic differential geometry makes the tangent functor representable. So what is the ultimately real higher-realm Platonic category of smooth spaces? Perhaps some sort of category of synthetic Frölicher spaces? Or maybe even synthetic diffeologies?
It turns out that people are thinking hard about such questions, and my imagination soars as a read the preprints of Hirokazu Nishimura on the topic. And surely, Andrew Stacey is working on similar things, and perhaps many others. Perhaps the true smooth world will reveal itself to mankind soon- and if so, how I would delight to lay eyes on it!
In the previous section, I revealed my discomfort with the definition of smooth manifolds. Do I find PL manifolds to be any better?
Let’s briefly recall the definition. An -simplex is the convex hull of points in which do not all lie on a common hyperplace. We say that is a polyhedron if, for each point , there exist a finite number of simplexes whose union contains a neighbourhood of . Polyhedron is said to be a PL manifold if, for every point , there exists a neighbourhood of and a PL homeomorphism . So a PL manifold is one which can be chopped up into simplices (higher dimensional generalizations of triangles), which makes the category of PL manifolds really discrete and combinatorial.
Any PL manifold has a triangulation (a decomposition into simplices); actually it has infinitely many of them, because any simplex can be chopped up into smaller simplices (a refinement of the triangulation). The Hauptvermutung, which for -manifolds is true for and false for , states that any two triangulations of a PL manifold share a common refinement. PL manifolds with a common refinement are said to be combinatorially equivalent.
Recall that the star of simplex is the union of all (closed) simplices containing . Given a triangulation, you can replace star by the cone over the boundary of centred at a point . Such a move is called a stellar move. Alexander’s theorem says that any two combinatorially equivalent triangulations of a PL manifold are related by a sequence of stellar moves and their inverses. Actually, each stellar move can be realized as a sequence of simpler “bistellar” or Pachner moves by a fundamental theorem of Pachner.
A PL manifold isn’t trying to be abstract- it’s a lego structure built up block by block, where the lego blocks are simplices. Alexander’s Theorem, and Pachner’s Theorem, then give us a framework in which to relate manifolds, and their invariants, which were constructed simplex by simplex.
This looks nice, but from the point of view of this post this is horrible- triangulations aren’t preserved under anything! Consider the product of two triangulated manifolds- say, the product of two line segments , each divided into subsegments . The square that is their product isn’t triangulated- instead, it’s partitioned into four smaller squares. Worse, the dual complex (-simplex for -simplex, -simplex between -simplices if corresponding n-simplices share an -face ) for an -manifold with boundary isn’t a triangulation. For Poincaré duality you need this complex, so obviously something is philosophically wrong.
So, along comes Whitehead, and he defines CW complexes, which are what we use when we teach courses on this stuff. Now, you’re gluing together closed balls instead of simplices. The maps are required to satisfy:
- The restriction of f to the interior of is a homeomorphism onto its image.
- The image under of is contained in the union of a finite number of elements of the partition, each having cell dimension less than .
In particular, you can connect blocks by some crazy continuous function (rather than just slotting together simplexes by the identity), and you can build all sorts of monstousities as CW complexes. From our perspective, though, the more serious problem is that there is no known analogue to Alexander’s Theorem, meaning that the CW decomposition is, in one sense, annoying extra data which we don’t really care about, and that if we use it to construct an invariant, we cannot then prove that the invariant is independent of the cellular decomposition.
People have realized that there is something amiss about this state of affairs at least since the 1950’s (Gugenheim). But, through inertia I suppose, it has never been rectified. One can get away with such things for quite a long time. Take products, then triangulate. Build as CW complexes, then triangulate. But in quantum topology, you are constructing invariants simplex by simplex, one simplex at a time, and having to work with huge numbers of simplexes and retriangulate every 5 seconds is seriously annoying. It’s actually even worse than that. If you want to construct extended TQFT, some of the most fundamental cells to consider are digons.
A paper I’m looking at, and which I’m currently quite excited about, is On piecewise linear cell decompositions by Alexander Kirillov. Motivated by (tremendously interesting) work with his student Benjamin Balsam (a must read paper, by the way), Kirillov defines a more modest generalization of simplicial decomposition, which he calls PLCW decomposition (I suggest PW decomposition as shorter, hence better). He defines a regular map as a PL map from convex compact polyhedron (a cell), such that the restriction of to the interior of is injective. In particular, can glue pieces of the boundary of together.
A PLCW complex is now defined inductively by starting with an -dimensional PLCW skeleton, and attaching -cells, such that the restriction to their boundary is a regular map. This is a simple and a natural enough condition that I can well-believe that such structures exist in some metaphysical sense. A product of PLCW decompositions is again a PLCW decomposition, and we have all of the cells that we want.
Another suggestion that PLCW is the way to go is the version of Alexander’s Theorem that is proven in the paper (I casually wonder about a Pachner Theorem). Stellar moves have natural analogues for PLCW complexes, and any two combinatorially equivalent PW (PLCW) decompositions are related by these moves.
The paper really is a joy to read, because it’s written so clearly, and the objects it considers are so intuitively natural. Why was it not written 70 years ago? I don’t know. But better late than never. The idea of PLCW decompositions is really cool, and I would love for it to enter into the topological mainstream!