Let me talk about the second chapter, “Topological Surfaces” in Bloch’s “A First Course in Geometric Topology and Differential Geometry”. I finished it quite a while ago, but I’ve had trouble deciding how to talk about it. I don’t want to just summarize it. Instead, I think I’ll try asking and answering some leading questions.
First of all, why did I choose to read Bloch? A few years ago, I was seriously shaken up by the following, from Freed & Uhlenbeck’s “Instantons and Four-Manifolds”, p. 1: “A basic problem is to ascertain when a topological manifold admits a PL [piecewise linear] structure and, if it does, whether there is also a compatible smooth [differential] structure. By the early 1950’s it was known that every topological manifold of dimension less than or equal to three admits a unique smooth structure.” They were setting us up for the fact that it isn’t generally true in dimensions greater than 3.
To be explicit about the challenge we face: topological, simplicial (which Bloch tells me generalizes to PL), and differential structures on a manifold coincide for low dimensions but not for high, and “high” means “greater than 3”.
This was all news to me. I knew, for a small value of knew, differential geometry, and I’d seen some simplicial stuff – triangulating and cutting up surfaces – and I was acquainted with topology, possibly for a large value of acquainted, but their “basic problem” was nothing I’d ever heard before.
Bloch is presenting separately the various structures for 2D manifolds – surfaces – even though they coincide for 2D. We can still keep the concepts separate. We will need to know what came from which structure when we move to 4D. I was ready for Bloch when I found him. (“When the student is ready, the teacher will appear.”)
What’s the big result in the chapter? The “classification theorem for compact connected surfaces (2D manifolds) in R^n”: any compact connected surface in R^n is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes.” Further, “the surfaces in this list are all distinct”. We could write
nT = T # T # … # T, a connected sum of tori;
nP = P # P # … # P, a connected sum of projective planes.
What’s the biggest disappointment in the chapter? That he doesn’t prove the classification theorem in this chapter. He does it at the end of the next chapter, after he’s introduced simplexes. Damn. There is a chance – skimming the proof – that he needs simplexes in order to prove that “the surfaces in this list are all distinct”. Either way, damn again. I was hoping for a proof that didn’t invoke simplicial structure.
What happened to the Klein bottle K? I don’t see it on that list. It’s there, as the connected sum of two projective planes: K = P # P = 2P.
What about the Möbius strip? It’s not a surface – because it has a boundary. Well, what’s a surface? it’s a subset of R^n every point of which has an open neighborhood that is homeomorphic to the open unit disk. No boundaries allowed.
What’s the big deal with boundaries? I’ve seen “manifolds with boundary”. Well, if we restrict ourselves to surfaces as being 2D manifolds without boundary, then, for example, it is true that every closed surface in R^3 (not in R^n) is orientable. Since the Möbius strip is a closed subset of R^3 but not orientable, better to not call it a surface. (We see something similar when we talk about unique factorization into primes: if we say that 1 is a prime, then 4 = 2×2 cannot be written uniquely as a product of primes, because we can put any number of 1s in the product.)
What’s the most fundamental theorem in the chapter? I have to name two: “schönflies theorem” and “invariance of domain”. Schönflies is just a tad more technical than I want to state here – but note that it is for R^2 not R^n; but invariance of domain is trivial to state: “if is homeomorphic to R^n, then U is open in R^n.”
What’s the big deal? A corollary is: if m and n are distinct integers, then R^m is not homeomorphic to R^n. (It’s the corollary which sounds more like “invariance of domain”.) We know that R^m and R^n are distinct vector spaces; now we know they are distinct topological spaces. There is another big deal to “invariance of domain”. In the differential geometry section of the book, I found: “Many differential geometry texts avoid using Invariance of Domain … by restricting to the use of coordinate patches that have continuous inverses.” Yes, I’ve seen that, they’re called “proper patches”. Now I know that they’re just a technical decision, not a fundamental one.
What’s the most satisfying theorem in the chapter? It’s the one that says a gluing scheme gives us a surface, and every surface has a gluing scheme. Hey what? A gluing scheme formalizes what we do when we take, for example, a square and imagine gluing its four sides by lining up the arrows and symbols. Here’s how we show gluing the sides of a square to get a torus:
The goal of describing surfaces as topological manifolds would have failed without this theorem, specifically, without a proof of this theorem that does not use simplicial structure. I can find plenty that do use it!
What was the cutest part of the chapter? (How’s that for an adjective applied to mathematics?) The connected sum # of compact connected surfaces is associative, commutative, and it has an identity (the sphere); that’s lovely, but what it doesn’t have is inverses! In fact, we can show that if A # B = S, then A = S and B = S (and I am using = to denote “homeomorphic to”); i.e. only the sphere S has an inverse, namely itself. (It must; after all, it is the identity.)
What did i like best about the chapter? The formalization of gluing schemes using identification spaces and quotient maps. Everyone shows me pictures of gluing schemes, but how do we describe them formally? OK, done.
What did I get out of other books? Having finished this chapter of Bloch, i picked up Firby & Gardiner’s “Surface Topology”, and Lee’s “Introduction to Topological Manifolds” (which has a chapter on surfaces).
From Firby & Gardiner, I got a more customary approach: a more casual approach to surfaces, and no distinction between topological and simplicial concepts. They had plenty of caveats and counterexamples, but they just didn’t prove a whole lot.
OTOH, they gave me a different statement of the classification theorem (they write the sphere as 0T, i.e. a torus with zero holes) :
an orientable compact surface is homeomorphic to nT, for some n >= 0.
a non-orientable compact surface is homeomorphic to
either nT # K
or nT # P, for some n >=0.
To show that this is equivalent to Bloch, we need to know that K # P = T # P, in addition to knowing K = P # P.
In Lee, I got to see the major simplification in Bloch. Where Bloch defines a surface as a subset of R^m which for which every point has an open neighborhood homeomorphic to the open unit disk, Lee gives us the more abstract definition. He defines an n-dimensional topological manifold as a Hausdorff second countable topological space which is locally euclidean of dimension n. Bloch gets the properties “Hausdorff second countable” by restricting to a subset of R^m; and, of course, he restricted to surfaces, n =2.
Lee describes the purpose of the two conditions as: Hausdorff tells us there are enough open sets; second countable tells us there aren’t too many of them.
I remind us that every abstract manifold, a la Lee, can be viewed as a manifold embedded in R^m for some m. While the abstraction may be useful – I’m sure it is – it gets us no new manifolds, none that we couldn’t have found by starting with subsets of R^m, a la Bloch.
Let me also remind us that the Klein bottle is a surface (a 2D manifold) but it sits in R^4, not R^3. That’s why I used both m and n in the previous paragraphs: for the Klein bottle, n = 2, m = 4.
Both Firby & Gardiner and Lee introduced the algebra of polygonal disks, typified by that square with arrows which I claim embodies a gluing scheme for the torus. Instead of the drawing, we could write a presentation
by starting at a, and moving CCW. We could relate operations on such “words” to geometric operations. Bloch did none of that.
We could also get the sphere, the projective plane P, and the Klein bottle in similar fashion:
Lee states the classification theorem as Bloch did, but he describes it using presentations; he also uses simplicial structure to prove it.
So, I’m disappointed that the proof of the classification theorem is in the chapter on simplicial structure, but for all I know, we wouldn’t have the classification theorem if we didn’t have a unique simplicial structure for any topological surface. Maybe there can be no proof without simplicial structure.
Still, it was good fun and I’m looking forward to chapter 3.