Simplicial curvature & simplicial Gauss-Bonnet

Introduction and text

The last section of Bloch’s chapter 3 (simplicial surfaces) is a long (and to my mind at this time, uninteresting) proof of the 2D Brouwer fixed point theorem: any continuous map from the disk to the disk has a fixed point. Bloch also proves a corollary, the no-retraction theorem, that there is no continuous map r from the disk to the circle such that r(x) = x for all x on the circle.

That one sounds interesting. We’ve seen in before, with the commentary that you can’t map the surface of a drum onto its rim without tearing it. I still don’t see it that way. But it is rather shocking that the map r cannot preserve all the points on the rim.

Anyway, we’re not going to fight with those. For me, the climax of chapter 3 is the simplicial Gauss-Bonnet theorem. It shows that there is a definition of curvature for simplicial surfaces (in fact, for polyhedra in general) such that the total curvature of a surface is equal to times its Euler characteristic .

(A simplicial surface is a polyhedron all of whose faces are triangles. I expect we’ll see this again in another post.)

That the total Gaussian curvature of a surface is equal to is called the Gauss-Bonnet theorem. It is a reasonable culmination of a first course in differential geometry. The simplicial version means that we have a definition of curvature for simplicial surfaces and polyhedra which gives us a form of the Gauss-Bonnet theorem. That says it’s a reasonable definition of curvature.

So what is this marvelous definition of simplicial curvature? It’s also called the angle defect, and goes back to Descartes.

Books Added: Differential Topology

Introduction

Putting out the following few books has been far harder than I expected, and has taken a lot more time. There are 6 of them: 3 texts, 1 reference, and 2 small sets of notes.

The fundamental problem is that I haven’t worked thru these books yet. Simply put, I’m effectively a grad student trying to figure out which books to read in order to introduce myself to a new field.

To put it more fancifully, I feel a bit like a wide-eyed urchin looking in a bakery window, trying to figure out what the different pastries will taste like, and I’ve picked out a few of them to try.

That simile fails, of course, because I’m not just looking at the pastries; I’ve held them in my hands and looked inside. I own these books, I’ve read each preface and table-of-contents, and I’ve read further into them. I’ve seen every one of them in other bibliographies; I’ve just read some of the reviews on Amazon….

The problem is, I haven’t gone into these books and come out the other side.

Nonplanar graphs & the Euler characteristic

Introduction and K4, the complete graph on 4 vertices

I want to show you something clever, but I’m going to omit the details of how we justify part of it. And I’m going to raise a question about another part of it. But I think this application of the Euler characteristic is interesting, even if I won’t or can’t cross all the t’s.

We can define the Euler characteristic of a graph as . We can show, for example, that every tree (a graph with no closed paths) has . If a graph is not a tree, then the closed path might create a face, but we don’t count the faces.

One question that arises when we have a graph is: is the graph planar? That is, can it be drawn in the plane so that edges do not have extraneous intersections?

Better to show you. Draw a square (or rectangle), and draw the two diagonals. This is called K4, the complete graph on 4 vertices, because every vertex is connected to every other vertex.

But the diagonals cross each other.

The Euler Characteristic: Teasers

These are things i came across when I first started looking at the Euler characteristic, in fact, when I was looking at triangulations in particular.

n-manifolds

The Euler characteristic generalizes to dimensions other than 2, and there are at least three noteworthy theorems involving the Euler characteristic. I’m not going to say much about them, because they, like so much else, are still outside my comfort zone. I’ll just barely tell you what they are, and leave you to chase them down if they interest you.

As we’ve seen, the Euler characteristic of a polyhedron is given by

,

where v, e, f are the numbers of vertices, edges, and faces. Homeomorphic polyhedra have the same Euler characteristic, and that means we can define the Euler characteristic of a topological surface as the Euler characteristic of any polyhedron which is homeomorphic to it.

This alternating-sign sum of the numbers of 0-, 1-, and 2- simplices generalizes in the obvious way: for an n-simplex, we take the sum, with alternating signs, of the numbers of k-simplices, for k <= n. As for surfaces, so for n-manifolds: this is a topological invariant, and we want to define the Euler characteristic of an n-manifold as the Euler characteristic of any k-simplex homeomorphic to it.

Topology: separation axioms

Introduction

I first came across the separation axioms in a functional analysis text (Bachman & Narici, “Functional Analysis”; Dover 1998, orig. 1966). I really like classification theorems, and these seemed really cool. As I said in the second post about general topology books, there is still not general agreement on the terminology. The mathematics is unambiguous, but there are two sets of intertwined terminology.

For example, the terms T4 and normal (to follow) are combined with the term T1 in either of two ways. T1 is unambiguous, but we either say that a topological space is

normal iff it is T1 and T4

or

T4 iff it is T1 and normal.

That is, there is a property called either T4 or normal. While we can study spaces which have that property alone, it is usually more interesting to study spaces which have that property and the T1 property. Such spaces are called normal or T4, respectively, depending on what name we assigned to the property. That’s the rub: is the property itself called T4 or normal? Then the other term is used for the combination with T1.

I choose to use the terminology typified by

Books Added: general topology 2

OK, I said I wouldn’t go buy more books because mine were old. I didn’t. I bought two more old books. I was looking on the internet for more about the “separation axioms” and came across these two. One was a familiar title that I probably should have gone looking for (“Counterexamples”), but I didn’t know of the other.

(Any discussion of the separation axioms must cope with the fact that there are two distinct sets of terminology. These books were cited as the epitomes of the two terminologies.)

They’re both very well reviewed and, it seems to me, excellent. Quite apart from that, they are also Dover paperbacks, which means they are quite affordable.

Willard is in the same class as Dugundji and Kelley: a textbook which is exhaustive enough to serve as a reference. Like Kelley, it has lots of problems, and many of them investigate auxiliary material. Oh, unlike the other two, Willard has a few pictures.

It is also fun to read. No, he’s not trying to be a stand-up comic, but every once in a while he phrases something nicely. “In the next (and obvious) step to normal spaces, we find ourselves confronted with the real bad boy among the separation axioms.”

Steen and Seebach is a compact presentation of topology (40 pages), beautifully organized counterexamples (120 pages), a summary of metrization theory (24 pages), and a collection of charts and tables for finding a desired example (20 pages). I would think, speaking as an onlooker, that this is an indispensable reference if you do much topology.

Need a reference text? Unless you need something specific from Dugundji or Kelley, I suggest you get Willard.

Doing topology beyond your first course? Get Steen & Seebach on general principles.

Books Added

Steen, Lynn Arthur and Seebach, J. Arthur Jr., Counterexamples in Topology, Dover, 1995 (orig. 1978),
ISBN 0 486 68735 X
[general topology; 17 Nov 2008]
Reference. Very well organized, with many charts of relationships.

Willard, Stephen. General Topology, Dover, 2004 (orig. 1970).
ISBN 0 486 43479 6.
[general topology; 17 Nov 2008]
Textbook and reference. Well-written. Copious historical references and notes.

books added: general topology (point set topology)

Introduction

Let me discuss my favorite general topology, i.e. “point set topology”, books. I have already discussed “algebraic topology” here.

Like so much other pure mathematics that I do not use professionally (for modeling power plants), topology is not on the tip of my tongue. But it’s fun, so I do it once in a while. And it’s fundamental, so I often have to go back to it when I’m playing with other mathematics.

This is a discouraging review in one respect: 4 of these 10 books are out of print: Kasriel, Dugundji, Sieradski, and Seifert & Threlfall. Heck, if you want Seifert & Threlfall, you should buy it in German! And for two of the books (Naber, Chinn & Steenrod) that Amazon claims to have in stock, there are multiple listings, many of which say the books are not available.

But I’m not going to go buy more books just because the ones I have are out of print. This is what I like, of what I have.

Buying Used Books

surfaces: visualizing the gluing of them

Quite some time ago, a friend asked me what would happen if we tried to construct a torus by gluing all 4 sides of a sheet of paper together, instead of first one pair then the other. Didn’t the math have to specify first one pair then the other?

One reason I’ve been hesitating over this post is that it doesn’t seem to be “real” mathematics – though any number of people might howl that PCA / FA isn’t “real” mathematics either. This is just a small drawing that I cobbled together to show that the homeomorphism between a circle and a line segment with endpoints identified… well, it doesn’t have to correspond to a physical process. (Why don’t I refer to “the glued line”?)

“… algebra provides rigor while geometry provides intuition.”
from the preface to “A Singular Introduction to Commutative Algebra” by Greuel & Pfister

It helped me to go back and read my original comment when I acklowledged Jim’s question to this post. I see that I did not understand that what matters to the formalism, the algebra, is before and after; what matters to the geometry is between or during. We reconcile them by permitting some things in the geometric visualization that we would not permit in the formal algebra, if the algebra even formalized the process: points passing thru points; or even some tearing, if when we reglue it we restore it rather than take the opportunity to change it.

That’s what bloch says, on p.57, discussing the physical process of getting from the knotted torus

to the regular torus.

The Euler characteristic (triangulations 2)

I want to talk about some things I saw when I was looking at triangulations. This also continues my reading in Bloch.

The major thing I saw was the Euler characteristic. We usually define it for polyhedra, as the alternating sum / difference of the number of vertices, edges, and faces (0-, 1- and 2- simplices)…

Then we would define it for a surface by taking the Euler characteristic of any polyhedron which is homeomorphic to that surface.

For that to be well-defined, of course, requires that all polyhedra which are homeomorphic to a surface S have the same Euler characteristic (as they do).

By the same token, we could define the Euler characteristic of a surface from any triangulation of the surface, after we show that all triangulations of a surface have the same Euler characteristic. Oh, and we’d better actually prove that every topological surface (every topological 2-manifold) can be triangulated.

Yes, they can be. Not true for topological 4-manifolds, and I think it’s still wide open for higher dimensions. In contrast, I believe that every differentiable n-manifold supports a unique piecewise linear (PL) structure (which is the generalization, they tell me, of triangulations).

Details.

Correction to Triangulations of Aug 12

I made a single large edit to the post, trying to clarify the Heawood theorem as opposed to the Heawood conjecture. This is mostly history of mathematics, but the distinction as I originally posted it was unreasonable.