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.

First off, it turns out that all the curvature of a polyhedron (or simplicial surface) is concentrated at the vertices. There is no contribution from the edges.

(Either recall or trust me that the Gaussian curvature of a cylinder is zero…. The simplicial curvature of an infinitely long polygonal cylinder ought to be zero. That is, if instead of a circular cross-section, our cylinder has a polygonal cross-section, we would like the total curvature to still be zero. I take it to be infinitely long so that it has no vertices, only edges and faces. Such a thing is not a compact surface, but it may help justify the idea that edges do not contribute to simplicial curvature.)

Imagine some polyhedron. There are at least two ways to describe the simplcial curvature. One is to look at each vertex. For a fixed vertex v, consider the angle at v for each of the faces which touch v, add up those angles, and subtract the sum from . (That’s why it’s also called the angle defect: it measures how far from , i.e. how far from planar, the polygons are when they’re touching. The pictures that follow may help.)

Note that we are looking at one angle from each polygon that touches the vertex v; we are not measuring an angle in space at the vertex. Anyway, add up all the angles at each vertex, and subtract that sum from , and call it the angle defect at v.

Then add up all the angle defects: that sum is the total simplicial curvature, and it’s equal to for that surface.

The other way to do it is to count all the angles first. Add up all the angles in all the faces. Subtract that sum from , where V is the number of vertices. (Each angle in a polygon is associated with one vertex.)

I believe I could write that compactly as:

Let K be a polyhedron, and let be a vertex. If is a face (polygon) containing v, let denote the angle at v in . The curvature of K at v (or the angle defect at v) is defined to be the number d(v) given by

where the are the faces of K containing v.

The total simplicial curvature of the polyhedron is the sum of the angle defects,

.

Let’s look at some examples.

regular tetrahedron

The simplest possible case is a regular tetrahedron: put 4 equilateral triangles together.

There are 4 vertices and every vertex is the same. At each vertex, 3 equilateral triangles meet, so the sum of the angles at one vertex is

so the angle defect at one vertex is

and there are 4 vertices, so the total simplicial curvature is .

By direct computation, the Euler characteristic of a tetrahedron is .

Therefore, we do indeed have

.

slighly irregular tetrahedron

What about an irregular tetrahedron? Suppose we put the vertices at the origin and on each axis at a distance of 1. There are 3 faces on each of the coordinate planes: these faces are right triangles. The 4th face is an equilateral triangle.

The 4th face does not touch the vertex at the origin (shown orange); we have 3 right angles there. The sum of the angles is ,

so the angle defect is

.

(Oh, of course! Look at the picture. More importantly, look at the open space at the orange vertex. That open space has an angle of . That’s where the comes from, and why we subtract from it.

At the other three vertices (shown yellow, blue, black; the black vertex is in 3 pieces), we have one equilateral face and two angles from the other two faces. At each of these 3 faces, the sum of angles is

,

so the angle defect is

.

If we look at either the yellow or blue vertices, the portion that is not faces is certainly larger than , and in fact we can see that it’s , which is, indeed, .

Now we add up all the angle defects

.

Once again, that’s .

What about a cube?

Looking at any of the vertices shown as blue dots, we should guess that the angle defect is at each of the 8 vertices, so the total is, once again,

.

Alternatively, 3 rectangles meet at each vertex, so the sum of the angles is

and the angle defect at each vertex is directly computed to be

.

OTOH, the Euler characteristic of a cube is

,

so .

(I know perfectly well that the cube is homeomorphic to the sphere, and therefore has the same Euler characteristic, but come on: we compute the Euler characteristics of polyhedra and simplicial surfaces, and then define the Euler characteristic of the sphere from them. I will admit, however, that if don’t get 2 by direct computation (for polyhedra without holes), I know I made a mistake!)

hexagonal cylinder (prism)

How about a finite polygonal cylinder? We know it’s homeomorphic to a sphere (), so we know the total simplicial curvature should be , but let’s work it out.

Here’s a regular hexagonal cylinder, or prism. There are 12 vertices; each is touched by 2 rectangles and a hexagon .

(Oh, cut the hexagon into 6 equilateral triangles; each angle at a vertex of the hexagon is made up of two angles.)

At each vertex, then, the sum of the angles is

,

so the angle defect is

and then the sum of those (i.e. 12 times the individual defect) is the simplicial curvature,

.

torus

How about a torus? First off, let’s grab a triangulation from here.

Just look at any interior vertex: there is no gap, no space where a face isn’t. The angle defect should be zero at every interior vertex.

But every vertex is interior. The ones that seem to be on the boundary are identified with those on another boundary.

Each of the angle defects is zero, the sum of the angle defects is zero, and we need

,

But that triangulation has 18 faces (that’s the easy number), 27 edges, and 9 vertices. .

Good. We’ve finally seen a case that didn’t add up to .

Gaussian curvature of the smooth torus

One last thing. The smooth torus, like the cylinder, has total gaussian curvature equal to zero. Unlike the cylinder, however, the torus does not have constant curvature equal to zero everywhere.

(When we take a finite cylinder and bend it to make a torus, we have to stretch the material that goes to the outside and compress the material that goes to the inside.)

In case you’re curious, if we parameterize a torus as

then I believe the Gaussian curvature (which I have not shwn you how to compute) is cos(u). (Really? Independent of the radii r and R? Yes! They show up in both the curvature and the element of area dA, and cancel out.) Integrated over , that will be zero. And zero integrated over will still be zero.

What did I actually get from that parameterization? (Take r = 1, R = 3.)

The red curve (outer rim) is u = 0, so cos u = 1 and the curvature > 0.
The black curve (inner rim) is , so cos u = -1 and the curvature < 0.
The green curve is , so cos u = 0 = curvature.

So we have seen some examples of computing the curvature of simplicial surfaces and polyhedra.

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.

That matters, because my opinion of books has been known to change after I finished them, though usually from bad to good. Sometimes what I thought was incomprehensible and poorly written ends up seeming a nice presentation after I have a global framework. (That was true of Dugundji, and of Ahlfors’ complex analysis, and of Dean’s elements of abstract algebra: they are all very nice books, looking back on them, now that I understand more about the subjects.) OTOH, I’ve also found books that seemed inviting and appropriate, only to run into insuperable obstacles. (Hamermesh’s group theory comes to mind: even today his book is still opaque to me; I have trouble following him even on things I understand.)

So, I am encouraged by the other reviews of these books, particularly in Bloch, on Amazon, and among themselves: Guillemin & Pollack recommend all but Hirsch – which hadn’t been published yet! Bloch recommends all but Guillemin & Pollack; Milnor Topology lists all but Hirsch and Guillemin & Pollack, which didn’t exist at the time.

book reviewers on Amazon

Let me point out that there are two especially detailed Amazon reviewers all of whose reviews seemed worth reading; there’s a button for doing that. One reviewer is “Malcolm”; he appears to be an expert in differential manifolds. In fact, I’ve just ordered two books based on his reviews. If you want to see his reviews, start with any of these four: Hirsch, Guillemin & Pollack, Munkres, or Milnor Topology. He did not review either Milnor Morse Theory or Wallace.

The other reviewer seems far more trustworthy than his name, “mathwonk”, might suggest. He’s a college math professor. To find him, look under the reviews of Wallace.

What is it?

So just what is differential topology? “That sounds rather formidable,” said a physicist friend yesterday.

If nothing else, it’s a very reasonable search string on Amazon.

Seriously, it ought to be the study of differentiable manifolds (smooth manifolds). But differential manifolds are, by intention, what we can do calculus on. And the calculus is a lot of machinery.

So I took every “differentiable manifolds” book off my shelves and went thru them, just to help put the differential topology books in perspective. And that was a lot of books.

One way to think of differential topology is: the topological properties rather than the calculus-related ones. More generally, one can think of it as differential manifolds without any additional structure “… such as lie groups, Riemannian manifolds, symplectic manifolds, vector bundles, foliations….” which Lee (I need to add his Smooth Manifolds to the bibliography) says he would call differential geometry.

More importantly, I have seen the subject, whatever it may be, referred to as “differential and algebraic topology” (by Jean Dieudonne). The math arxiv (front end) lists general topology, algebraic topology, and geometric topology, and the last includes “manifolds”. Bloch says that geometric topology more narrowly defined would be topological and piecewise linear manifolds, but not differential.

I think five of these six books can be characterized as: the topological properties of differentiable manifolds without much regard to additional structure, and without using algebraic topology (that’s important for these books if not for the subject).

And I’m in trouble already. These books are going to discuss vector fields. Surely that counts as additional structure. Well, that’s why I allowed for some regard of additional structure. The fact remains that these books look different from my differentiable manifolds books, which in turn look different from my differential geometry books. I hope that as I add more books to the bibliography, I can make some sensible distinctions.

Overview

I started out with 6 books in hand. I wondered if I should add some differential manifolds books; I wondered if I should drop the Milnor “Morse theory”. In the end, I stayed with the original 6, having seen no compelling reason to add or subtract from that list.

Ah, one last point. Munkres specifically uses the phrase and title “elementary differential topology” to mean those theorems in differential topology which do not require algebraic topology for their proofs; similarly, as he says, a theorem in number theory is said to be elementary if its proof does not involve functions of a complex variable. (And that doesn’t mean the proof is easy.)

Oh, another last point. Guillemin & Pollack use an interesting rating scale, for difficulty. There is a perfectly straightforward one used by the MMA (Mathematical Association of America). Since North America uses phrases like “grade 10″, the MAA uses numerical grade levels, extending grades 1-12 up to 18 for “2nd year graduate studies”. A book suitable for seniors and first-year graduate students, then, would be marked “16-17″.

(A very little kid asked me what grade I was in in school, when I was a college freshman. Turned out, he didn’t recognize the word “freshman”, so I told him I was in the 13th grade.)

Guillemin & Pollack took another rating scale that we are familiar with: movies. G = elementary, with only analysis and linear algebra as prerequisites. PG means it requires something more, whether that’s some abstract algebra, or some topology, or whatever. R denotes graduate level mathematics, and X would be “hard going for a graduate student, meant to be read more for inspiration than for comprehension.”

I doubt that I will use either of those systems. I will comment that Guillemin & Pollack labeled Milnor Morse Theory and Wallace as PG; they suggested Milnor Topology as collateral reading with them, but did not rate it. Since Guillemin & Pollack was designed as a “leisurely first year graduate course” predominantly accessible to juniors and seniors, I would guess Guillemin & Pollack and Milnor Topology would have PG ratings.

Ok, on to the books.

Munkres is a reference book. It’s only 109 pages. Part I proves “the folklore theorems” of the subject: those things that almost everyone else says “go look somewhere else for a proof”. Skimming it, I get the impression that it would be a good place to start sharpening ones tools by working out the exercises (and Malcolm’s review concurs). Part II proves that every differentiable manifold has a unique smooth triangulation. That’s pretty major. Both Milnor Topology and Wallace cite this book.

I’ve repeatedly said that TOP, PL, and DIFF (topological, piecewise linear, and differentiable) structures are not equivalent in higher dimensions. And I also don’t understand the distinction between simplicial, piecewise linear, and triangulations.

Munkres gives me a precise definition of triangulation as he used it for the theorem. That’s a big step.

Wallace is where I want to start. I’m not sure I’ll make it all the way, because his chapter on “spherical modifications” looks rather intimidating. But Wallace seems to have been a significant researcher in the great breakthroughs of the 50’s and 60’s, and – more importantly for a first text – he is a good writer and expositor. (I own two other books by him.) More importantly for the subject, Wallace culminates in a proof of the classification theorem for compact connected smooth surfaces – but using critical points of functions on manifolds (this is the beginnings of Morse theory, I think) instead of using simplicial complexes. Since it’s a Dover paperback, you could buy it just for the classification theorem, and you shouldn’t care that it’s only 130 pages. Everyone else I’ve been reading proves the classification theorem for triangulable surfaces. (Ok, I haven’t really been reading Hirsch, but he also does it using the differentiable structure rather than simplicial; but I’ll get there sooner in Wallace.)

So I’ll be picking up Wallace first.

But he’s a special purpose text: one might say, a set of notes culminating in an important theorem.

Guillemin & Pollack is a full-blown text, aimed at graduate students but much of it accessible to undergraduates. Its four main sections are manifolds & smooth maps, transversality & intersection, oriented intersection theory, integration on manifolds. Within them, it lists immersions, submersions, and embeddings; transversality, intersection theory mod 2, and winding numbers; the poincare-hopf theorem, the Euler characteristic, and triangulations; the gauss-bonnet theorem.

I’ll be picking up this after Wallace. (Sadly, Guillemin & Pollack is out of print; the other 5 are still in print.)

Well, not quite. I’m going to do something a little strange, even for me.

Both Guillemin & Pollack and Hirsch are introductions to the subject, aimed at beginning graduate students. Guillemin & Pollack is the more elementary of the two, but Hirsch has a considerable amount of discussion.

I say, a considerable amount of discussion. Hirsch appears to be more detailed, and more advanced, so I would work thru it after Guillemin & Pollack. but I intend to read Hirsch first, and take notes on his remarks, before I start working thru Guillemin & Pollack. That’s how insightful his commentary seems.

Then I can work thru Hirsch. It looks more like an introduction to the key non-algebraic techniques of the subject: transversality (of course); vector bundles and tubular neighborhoods; degrees, intersection numbers, and the Euler characteristic; Morse theory; cobordism; isotopy; and he closes with the classification of compact surfaces using Morse theory.

This brings us to the two Milnor books. Milnor won the fields medal in 1962, and for that alone I’m very interested in what he chooses to say about a subject. Furthermore, he is famous for his lectures. The beginning of the Topology looks a lot like the beginning of Wallace, so I’ll keep it handy from the get-go. Heck, I should quote Malcolm’s review on Amazon: “… too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.”

The best I can do for the Morse theory is to quote an unknown reviewer (“A Customer”, which appears to be the generic name) from Amazon. He said that his advisor gave him this book, saying, “You’re not ready for this yet, but you should have it — it’s the best piece of mathematical exposition there is.” I am pretty damned sure that any bibliography of differential topology is flawed which does not include this book.

In summary, then, 3 textbooks (which I will do in order Wallace – Guillemin & Pollack – Hirsch), 1 reference (Munkres), 1 collateral reading (Milnor Topology), and 1 for dessert (Milnor Morse Theory).

The Books Added

Wallace, Andrew H. Differential Topology: First Steps. Dover 2006 (orig 1979).
ISBN 0 486 45317 0.
[differential topology; 20 Dec 2008]
Looks like a marvelous undergraduate introduction. From the preface: “… in this field, as indeed in any branch of topology, the first steps should be geometric.” He uses differential methods to obtain the classification of 2D manifolds. Short intense guide to further reading. Structured exercises, but no solutions. Epilog.

Hirsch, Morris W. Differential Topology. Springer, 1991 (corrected 4th printing).
ISBN 0 387 90148 5.
[differential topology; 20 Dec 2008]
“Graduate Texts in Mathematics”. The discussion and remarks throughout the text are worth reading first, in my opinion. Uses Morse Theory to get the classification theorem for surfaces.

Munkres, James R. Elementary Differential Topology. Princeton, 1966 (revised ed.).
[differential topology; 20 Dec 2008]
Reference. Proves the “folklore theorems”; culminates in the proof that any smooth manifold has a smooth triangulation.

Milnor, John W. Topology from the Differentiable Viewpoint. Princeton, 1997 (orig 1965).
ISBN 0 691 04833 9.
[differential topology; 20 Dec 2008]
Notes on a few selected topics. Legendary might be more appropriate than “a classic”. Brief guide to further reading.

Milnor, John W. Morse Theory. Princeton, 1969 (orig 1963).
ISBN 0 691 08008 9.
[differential topology; 20 Dec 2008]
A compact presentation, but also legendary. I think of it as dessert after Hirsch.

Guillemin, Victor; Pollack, Alan. Differential Topology. Prentice Hall, 1976.
ISBN 0 13 212605 2.
[differential topology; 20 Dec 2008]
Graduate / advanced undergraduate. A “leisurely first year graduate course”, predominantly accessible to juniors and seniors. Guide to further reading.

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.

But they don’t have to. Take one of them and stretch it, and pull it outside the square.

K4 is called a planar graph, because its edges can be laid out in the plane so that they do not cross. I might say that it’s manifestly planar. Its Euler characteristic is .

Although it has 3 faces, we aren’t counting them. (OTOH, it’s not an accident that it has 3 faces and its Euler characteristic is 1 – 3; I think each loop added to a tree subtracts 1 from the Euler characteristic.)

Now, the theorem I’m not going to prove (even Firby & Gardiner don’t prove all of it), is that if we embed a planar graph onto the sphere, and then count faces created by the embedding, the embedded graph has the Euler characteristic of the sphere:

We’re going to almost show that a couple of graphs cannot be planar; we will use to compute the number of faces created by the embedding, and we will show that either

is false.

Got that? I’m not going to prove that the Euler characteristic of the embedded graph is 2, but I’m going to use that result to compute the number of faces of a planar graph.

Why do I say embedding? Let’s look at K4, the square with diagonals: we have v = 4, e = 6, and we infer from that the embedded graph has 4 faces.

But when we drew the graph, it had 3 faces. Ah, but on the sphere there’s all the rest of the sphere, the “ocean” if you will. Note that the “ocean” touches 3 edges when I draw the graph so that it is manifestly planar. We’ve added a face when we embed the graph on the sphere.

So the Euler characteristic of a graph doesn’t count any faces; the Euler characteristic of a graph embedded on a surface does count faces.

Just for the fun of it: for K4, 2 e = 12, 3 f = 12, and we do have . In fact we have equality because every face has exactly 3 edges.

K5, the complete graph on 5 vertices

That was K4. Here is K5, i.e. 5 vertices and they’re each connected once to every other vertex:

Is it planar?

Embed it on the sphere. We have 5 vertices by definition, and (1/2) (5) (4) = 10 edges (“5 choose 2, order doesn’t matter”) From the Euler characteristic , we infer that the embedded graph has 7 faces. To be clear, if the graph K5 is planar, then the embedded graph has Euler characteristic 2 and 7 faces.

But we have

2 e = 20

and 3 f = 21,

and therefore 2 e < 3 f instead of .

We conclude that K5 cannot be planar.

(That inequality assumes that very face has at least 3 edges. Well, it can’t very well be a face with only 2 edges, can it? Can it?)

K3,3, the complete bipartite graph on 3 & 3 vertices

What about K3,3, the complete bipartite graph on 3 & 3 vertices? This is also called the utilities graph: we may imagine we have 3 utilities (gas, water, electric) and 3 houses, and every house needs to be connected to every utility. It’s called bipartite because there are two disjoint subsets of vertices, and all the vertices in one subset are connected to all the vertices in the second subset (and vice versa).

Is it planar? Can we lay out the utilities all on one level without the lines bumping into each other? (Of course, my electricity comes in on overhead wires, but in principle the town could have required that the electric line be underground, as the gas and water are.)

Here’s the graph:

Let’s check the same inequality, . We have 6 vertices and 9 edges. If the graph K3,3 is planar, then the embedded graph has Euler characteristic 2 and 5 faces… then 2e is 18, and 3f is 15, and the inequality is satisfied:

.

That tells us nothing.

Let’s think about it some more. Start at a vertex (“a”), follow an edge to a vertex (“b”) (which must be on the “other side”). If we return on the same edge, we don’t get a face. If we return on a different edge, we end up at a different vertex “c”. To get back to “a”, we have to go to the other side and back again.

That is, each face (except possible the “ocean”) has at least 4 edges. Let’s not worry about the ocean for now. If f3 = 0 (“no faces have 3 edges”), then we can show that

.

(The key is that every edge must separate two faces; then if every face has at least three edges,

where is the number of faces having i edges.

Now set f3 = 0, and replace :

since )

So we compute

2 e = 18 < 20 = 4 f,

instead of ,

and we conclude that K3,3 is not planar… except that we really need to argue that the ocean touches at least 4 edges.

Do we? If we suppose the ocean only touches 3 edges (i.e. f3 = 1), we can show that

.

But, instead, we have

2 e = 18 < 19 = 4 f – 1.

We still have a contradiction, so K3,3 is not planar.

It turns out, that’s all folks. Kuratowski’s theorem (1930) says that a graph is not planar if and only if it contains K5 or K3,3. They are the “atoms” of nonplanar graphs; at least one of them is contained in every nonplanar graph.

a distressing observation

Having done that, let me back up and confuse you. Let me show you the t’s I’m not crossing. (In addition to not proving that the Euler characteristic of the graph embedded on the sphere is 2.)

Let’s take a planar graph with 2 vertices and 1 edge, i.e. a single line segment. It’s the tree with 2 vertices. Embed it on the sphere, and infer that it has one face.

The ocean.

Which touches one edge.

Now, the ocean really seems to be a face, i.e. a 2D thing. But it only has 1 edge. Grrr. Did you believe me when I said that something couldn’t be a face if it only had two edges? That’s OK, I believed me, too.

Oh, don’t seriously think of checking

.

That came from the assumptions that every face has at least 3 edges (false), and every edge touches 2 faces (false, but close: in this case our single edge does have “ocean” on both sides).

We could also take a tree on 3 vertices, i.e. the previous graph with a vertex added inside the edge. Embed it on the sphere. Now we have 2 edges, 3 vertices, and we compute that it has 1 face, and that face touches all 3 vertices and both edges. But that face does not have 3 edges. I mention this case because the books limit the original theorem (“the Euler characteristic of a planar graph embedded on the sphere is 2″) to graphs with at least 3 vertices, which seems to include this case.

My objection in both of these cases is that the ocean is not a nice face. What we have not done for K5 or K3,3 is to show that there isn’t some weird shape of ocean that makes them planar but does not force the inequalities which we know are false.

Clearly, we would have to do some more work to make all of this hang together properly. What we’ve got is two really nice plausibility arguments that K5 and K3,3 are not planar.

When I check the graph theory books, they all seem to sail right past the number of edges on the ocean, and the requirement that every edge separates two faces. (Again, for these two trees, each edge does have a face on each side, but it’s the same face!)

References

Every one of the four graph theory books in the bibliography (Bollobas, both Bondy & Murty books, and Biggs, et al.) proves that K5 is nonplanar because it violates ; and also that K3,3 is nonplanar because it violates . Only Bollobas does not prove Kuratowski’s theorem. Biggs et al. contains Kuratowski’s paper.

Both Bondy & Murty books also prove that K5 and K3,3 are nonplanar because they violate the Jordan Curve theorem.

Finally, Firby & Gardiner, and Sieradski, prove that K5 is nonplanar because it violates , and they leave K3,3 to the exercises, without a hint.

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.

But we’ve been told that topological, simplicial, and differential structures on manifolds do not always coincide for n > 3. We can define the Euler characteristic for k-simplices and for manifolds with simplicial structure, but I am going to look skeptically at the Euler characteristic of topological and differentiable manifolds until I learn more about the different structures.

BTW, we can go down a dimension, too: we can define the Euler characteristic of a graph as v – e.

Betti numbers

The Euler characteristic of an n-manifold has an extremely interesting equivalence: not only does the formula generalize from 2 to n, but it can also be generalized to a topological space X: we can write

where is the kth Betti number of X, and we assume that eventually (i.e. for k large enough) . (Lee, “Introduction to Topological Manifolds”; or Rotman, “An Introduction to Algebraic Topology”.)

But hold on just a minute. The Euler characteristic was defined for things with simplicial structure, and the Betti numbers are defined (not by me!) for topological spaces. But we know that simplicial & topological do not coincide for higher dimensions than 3, so we probably have to be rather careful here.

When all else fails, look carefully at the book. What we actually have is two definitions, a theorem, and an extension of the definition. We define the Euler characteristic of a simplicial complex, and the Betti numbers of a topological space; the theorem says that for a finite simplicial complex, the Euler characteristic is equal to the alternating sum of the Betti numbers:

Having the theorem, we then extend the definition of the Euler characteristic to any topological space: we define the Euler characteristic as the alternating sum of the Betti numbers; the theorem assures us that the two definitions of the Euler characteristic coincide on simplicial complexes.

We’re still not out of the woods. Looking in Rotman, I am reminded that there are different homology theories. Do they all give the same Betti numbers? I have no idea. (Well, I suppose they do, but I’m ready for anything.) Gee, no wonder algebraic topology is still on my list of things to study.

As an aside, in Massey’s combined book, “A Basic Course in Algebraic Topology”, I find a note which says that until about 1930, mathematicians focused on the Betti numbers (and the torsion coefficients) of homology groups, rather than on the groups themselves.

Index of a surface

Next, on this detour, we have the Poincare-Hopf index theorem, which says that the Euler characteristic of a surface is equal to the index of the surface.

Hey what?

As the Betti numbers came out of left field, the study of vector fields on a surface is coming out of right field. If you’ve ever been told that you can’t comb the hair on a ball, you’ve encountered one of the key theorems of the subject.

There are two ways to look at the index theorem. In the special setting of dynamical systems (i.e. differential equations), we are looking at points where the vector field vanishes (critical points, which are equilibrium points, but not necessarily stable equilibria). More specifically, we are interested in the flow pattern in the vicinity of critical points.

In this context, we break flow patterns into sectors each of which is called parabolic, hyperbolic, or elliptic; and we have a recipe (due to Bendixson) for planar systems which says that the index of a critical point (wrt the vector field) is

I = 1 + (e-h)/2,

where e and h are the numbers of elliptic and hyperbolic sectors. (The parabolic sectors don’t contribute to the index.)

The index of the surface wrt the vector field is the sum of the indices of the critical points. The Poincare-Hopf theorem says that the

index of the surface (in fact, of an n-manifold) wrt the vector field is equal to the Euler characteristic of the manifold.

(I believe that Poincare proved it for surfaces and Hopf extended it to higher dimensions. I am not sure that the definition of sectors applies in higher dimensions; that is, I am not sure how the index is defined in higher dimensions.)

This imposes significant constraints on the possible vector fields.

That’s one way to do it. My references for this approach are Perko, “Differential Equations and Dynamical Systems” and Firby & Gardiner, “Surface Topology”.

I was hoping to at least construct illustrations of the three kinds of sectors, but I’m finding it difficult and very annoying to update my version 5 drawings to version 6 of Mathematica®. (Backwards compatibility? What’s that?)

From Firby & Gardiner, I extract the following description. A sector is parabolic if all paths lead to the critical point, or all paths lead away. A sector is elliptic if all paths begin and end at the critical point. A sector is hyperbolic if all paths sweep past the critical point.

But there is another way to define and compute the index of a critical point. In fact, we can compute the index of any point wrt a curve, but if it’s not a critical point, then its index is zero; therefore we can confine our attention to critical points. This approach starts with winding numbers. Fulton, “Algebraic Topology: A First Course”, is excellent for surfaces (with multiple chapters on winding numbers and vector fields), and even suggests how to generalize to higher dimensions. Also, Sieradski, “An Introduction to Topology and Homotopy”, has a few sections, rather than chapters, on winding numbers and vector fields.

I’m beginning to look forward to tackling Fulton.

Gauss-Bonnet

The Gauss-Bonnet theorem, as O’Neill states it (“Elementary Differential Geometry”) says that the gaussian curvature, integrated over all of a compact orientable surface, is equal to , where is the Euler characteristic of the surface.

Oh, but doesn’t that require differentiable structure?

Yes and no.

Bloch shows us the Gauss-Bonnet theorem for simplicial surfaces! It turns out we can associate curvature with the vertices of a simplicial surface, in such a way that the total curvature (the sum of the curvatures at each vertex) is equal to .

I’ll talk about this when I discuss chapter 3 of Bloch. I finished the chapter eons ago, but I haven’t tallied it up yet.

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

a topological space is normal iff it is T1 and T4

and I will explain why after we’ve seen the definitions. (This is the terminology of Steen & Seebach, Munkres, and Sieradski. The other terminology is used by Willard, Kasriel, McCarty, Kelley, and Adamson; Dugundji assumes that all topological spaces are T2 = Hausdorff, in which case the two terminologies coincide.) Based on simply counting authors on each side, I should go with the other convention. But I’m going to go with my favorite topology author, Munkres, and I have another reason for my choice.

There is an interesting wiki article about the history of the terminology here. This is what sent me off to get Steen & Seebach and Willard. To judge from the wiki article, the terminology I have chosen is old-fashioned, but not entirely out of fashion. These two books do, in fact, seem to epitomize the two alternative terminologies, because they are both thorough and general.

Let me remind you that a topology is a very general object. Willard puts it very nicely (p. 23):

“The ‘reasonableness’ of the following definition… can best be justified only by reading the forty-two sections which follow it.”

Given a set X, any collection of subsets T (called open sets) is called a topology for X if

• the empty set , and X are in T
• the arbitrary union of elements of T is in T
• the finite intersection of elements of T is in T.

The pair (X, T) is called a topological space.

The two extreme examples of a topology are

• the discrete: T = {every subset of X} = the power set of X.
• the indiscrete: .

(So the discrete topology is the largest possible topology on a set X and the indiscrete is the smallest possible.)

An intermediate example is:

Let X = {a, b, c}, a set of 3 elements. Let . That is, the singleton {a} is open but {b} and {c} are not; {b, c} is open but neither {a, b} nor {a, c} is open.

The point is that a topology is any collection we want, subject only to the three somewhat loose requirements.

I will also use that intermediate example later.

For his introduction to the separation axioms, Willard is again worth quoting (p. 85):

“Our definition of a topology admits structures which are, for most purposes, useless.”

Considering that Dugundji assumes that all spaces are Hausdorff (T2), I used to think that the definition of a topological space was a case – a rare case – of too much generalization, of a definition beautiful and simple but too general to be useful. There are, however, at least two kinds of non-Hausdorff topological spaces: the Zariski topology on an algebraic variety (it is T1), and the quotient topology on a space of equivalence classes (it need not even be T0). (Folland, “Real Analysis: Modern Techniques and their Applications; Wiley Interscience 1999, 2nd ed.)

First we look at separating points by means of open sets. (The indiscrete topology has the defect that each singleton set or “point” of X is contained in exactly one open set, X itself. We can’t distinguish any points from each other by using the open sets, i.e. by using the topology. Hence the name.)

T0 property and spaces

A topological space X has the T0 property if there exists an open set which separates any two distinct points: if x and y are distinct points of X, there exist an open set which contains one but not the other.

Let me be more explicit. A topological space X has the T0 property if, for any two distinct points x and y in X, either there exists an open set M(x) containing x which does not contain y, or there exists an open set N(y) containing y which does not contain x.

NOTE that the space X is an open set containing x, but it contains y, and vice versa. We are asserting the existence of a smaller open set, but not necessarily for both points of the pair. If we assert that both M(x) and N(y) exist, that’s the next property.

Here’s a picture of T0, showing an open set containing y that does not contain x. A T0 space is sometimes, but rarely I think, called Kolmogorov.

T1 property and spaces

A topological space X has the T1 property if x and y are distinct points of X, there exists an open set M(x) which contains x but not y, and an open set N(y) which contains y but not x.

One crucial property of a T1 space is that points (singleton sets) are closed.

This time each point has an open set which contains it but not the other. NOTE that we did not assert that the two open sets do not intersect, merely that their intersection contains neither x nor y. (That’s the next property.) Here’s a picture of T1, showing open sets which intersect, but their intersection, as we require, does not contain x or y. A T1 space is sometimes, but again rarely, I think, called Frechet.

T2 property and spaces

A topological space X has the T2 property if x and y are distinct points of X, there exist disjoint open sets M(x) and N(y) containing x and y respectively. Here’s a picture of T2. A T2 space is almost always, in my experience, called Hausdorff.

One crucial property of a Hausdorff space is that limit points are unique. (No, I haven’t defined a limit point. That’s another interesting subject.)

T2 1/2 property and spaces

A topological space X has the T2 1/2 property if there exist open sets whose closures are disjoint, which contain any two distinct points: for any distinct points x and y, there exist open sets A = M(x) and B = N(y) such that . I have seen this called either completely Hausdorff (Steen & Seebach) or Urysohn (Willard); the problem is that both those names are used for two different things (see below). I have only seen the “T2 1/2″ used to denote the property I have defined, but I’m not that widely read. Because of the inconsistency, I’m going to refer to T2 1/2 in what follows, although this does not seem to be widely used. I may adopt it permanently for myself, until and unless it leads to difficulties.

Here’s a picture, using solid lines to indicate the closures of the open sets containing x and y.

To this point, we have that

.

Life is good.

T3 and regular

Now we look at separating sets instead of points, still separating them by open sets of some kind.

First we separate a point and a closed set. (A set A in X is closed if its complement X – A is open; the closure of A, , is the smallest closed set containing A.)

A topological space X has the T3 property if there exist disjoint open sets which contain any closed set and any point not in the set: for any closed set B and any point , there exist disjoint open sets containing x and B respectively.

Here’s T3. This time I use uppercase (“B”) and color to denote the closed set.

It is crucial that the following set and topology (shown earlier as “an intermediate example”) is T3 but not T1 (the problem is that the point a is not closed):

X = {a, b, c}
.

This is why and where we need to combine properties in order to get especially worthwhile topological spaces. (Yes, we can study T3, T4, and T5 spaces per se. it is more fruitful to study T3 + T1, T4 + T1, and T5 + T1.)

We say that a space is regular if it is T1 and T3.

(In fact, we can show that if a space is T0 and T3, then it is T2, hence T1, hence T1 and T3. this means we could have defined a space as regular if it is T0 and T3. Of course, T1 and T3 immediately implies T0 and T3, so the two possible definitions of “regular” are equivalent.)

Although I used “normal” and “T4″ in the introductory discussion, the alternative terminology appears here as well. It applies to all subscripts 3 and higher. Where I say that a topological space is

regular iff it is T1 and T3

other people use regular to refer to my T3 property, and say a topological space is

T3 iff T1 and regular.

Whereas the progression of the earlier separation axioms kept tightening the requirements on the open sets whose existence we asserted, here we just replaced a point by a closed set. That would be a refinement of the earlier property if points themselves were closed sets. But that’s T1, and that’s why we want to study spaces which are both T1 and T3.

T4 and normal

Now we separate two closed sets instead of a point and a closed set. A topological space X has the T4 property if there exist disjoint open sets which contain any two disjoint closed sets: for any disjoint closed sets A and B, there exist disjoint open sets containing A and B respectively.

I should mention that a bad property of T4 spaces is that T4 is not hereditary: not every subspace of T4 is T4.

We say that a space is normal if it is T1 and T4. We still have the analogous: not very subspace of a normal space is normal.

T5 and completely normal

Two subsets A and B of topological space are separated if

.

A topological space X has the T5 property if there exist disjoint open sets which contain any two separated sets: for any separated sets A and B, there exist disjoint open sets containing A and B respectively.

I should mention that an alternative equivalent definition of T5 is that: a space is T5 iff every subspace is T4. It corrects the problem with T4.

We say that a space is completely normal if it is T5 and T1. We have the analogous: a space is completely normal iff every subspace is normal. It corrects the problem with normal, too.

Consider the two open intervals A = (0, 1/2) and B = (1/2,1) with the usual topology of the real line. The sets do not intersect: ,, but the closed intervals, their closures, do:


and .
Nevertheless, A and B are separated, because .

A and B have the T5 property because A and B themselves are disjoint open sets.

All of those properties, T0 thru T5, asserted the existence of open sets, sometimes satisfying additional conditions.

T3 1/2 and completely regular

We have an intermediate property which is described differently.

Given two disjoint subsets A and B of a space X, a Urysohn function for A and B is a continuous function such that f(A) = 0 and f(B) = 1.

Urysohn’s Lemma, then, says that if A and B are disjoint closed subsets of a T4 space, then there exists a Urysohn function for A and B.

A topological space X has the T3 1/2 property if there exist a real-valued continuous function which separates an open set from any point not in it: i.e. for each open set and each x not in U, there exist a Urysohn function f for x and U.

We say that a space is completely regular (or Tychonoff) if it is T3 1/2 and T1.

Urysohn function for distinct points

We could, instead, require this kind of property for two points: we would say that a space is Urysohn if it has a Urysohn function for any two distinct points.

Implications of the properties

At this point, thanks to adding T1 to the definitions, we can show (!)

completely normal normal completely regular regular T2 1/2 T2 T1 T0.

The implications among the Ti properties (for i > 2 1/2) are not so pretty.

Note that a Urysohn space was not in that list. Instead of the subsequence

completely regular regular T2 1/2

we could have written

completely regular Urysohn T2 1/2.

but there is no inclusion relationship between Urysohn and regular. We have two beautiful inclusions, if we omit either regular or Urysohn, but not if we include both.

This is the second reason why I decided to follow Steen & Seebach and use T’s for the properties and names for the combinations. If we did it the other way, with names for the properties and T’s for the combinations, we could write

T5 T4 T31/2 T3 T2 1/2 T2 T1 T0,

or, more elegantly,

Ti Tj for i > j, with i, j in {0,1,2,21/2,3, 31/2,4,5}

but then we’ve left Urysohn spaces out in the cold. Since the theorem is no longer pretty, I chose to use the shorter Ti to denote a property, and write, for example,

normal = T1 + T4.

I first saw them the other way:

T4 = normal + T1, etc.

and it is possible that I would not have been so struck by them without the lovely Ti Tj for i > j. (Adamson emphasizes that he chooses this convention because of the simplicity of that statement.) Nevertheless, I have presented them the other way.

The fact is, if you’re studying someone else’s work, you may have to adopt their terminology as long as you’re there.

One final caveat: completely hausdorff and Urysohn

Willard and Steen & Seebach use exactly opposite definitions for completely Hausdorff and Urysohn. Willard says a space is completely Hausdorff if any two distinct points have a Urysohn function, while a space is Urysohn if any two distinct points are separated by open sets whose closures are disjoint (T2 1/2).

It seems odd to say that a completely Hausdorff space is separated by a Urysohn function, while a Urysohn space is defined by separation by sets.

Note that this naming mess is between Urysohn and completely Hausdorff, while the two spaces that mess up our nice inclusion are regular (“T 2 1/2″) and Urysohn (“two distinct points have a Urysohn function”). Note, also, that unlike the confusion between T4 and normal, this mess has nothing to do with T1.

My own inclination, on this, is to use T2 1/2 for the space defined by separation by sets, and Urysohn for the space defined by “two distinct points have a Urysohn function”.

Ah, let me close with a few remarks. First, every metric space – with the usual topology defined by open balls – is completely normal (T1 + T5). Note that I have specified my naming convention. The implication does not go the other way: normality is not sufficient to imply metrizability (that the topology can be gotten from a metric on X).

Second, there are a host of fascinating properties which I have not mentioned: first and second countable, first and second category, assorted forms of connectedness, and assorted forms of compactness.

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

Let me point out that I buy used books online via abebooks.com. I have no connection with them except as a mostly satisfied customer. I think I have bought 10-15 books from them, and only one was not as advertised: it did not have “clean unmarked pages”. OTOH, it was so rare and hard to come by that I chose to keep it anyway: the student who marked it up stopped after page 51 and the rest of the book, the good stuff, was clear. (I could have returned it at the bookstore’s expense with a full refund, but I chose not to.)

I also buy used books online from powells.com; it’s a huge physical complex of bookstores (it really is huge!) in Portland Oregon, and wandering through them is almost a religious experience for bibliophiles. I have no connection with them except as a completely satisfied customer. Abebooks includes books available from Powell’s, but I buy such books directly from Powell’s because I’ve been there and I like them.

Two First Courses

First, there are two fine textbooks, intended and quite suitable for, a first course. Munkres came highly recommended, and from my own reading of it, I would imagine that it is the standard text. Well-written, with lots of examples. Good enough that I bought everything else he wrote; fortunately for my wallet, that’s only 3 other books, one of which is already in the bibliography.

I have no idea how I got the other book, by Sieradski. It is used, worn but unmarked. I suspect I bought it in person, but I have no recollection of where. It’s a little denser than Munkres, by which I mean only that it seems to have fewer words between definitions and theorems, but it still has plenty of descriptive text, and it has lots of examples and drawings.

If I feel like reading topology, I grab both of these and see what catches my fancy.

Sieradski is out of print, but if you’re interested in topology and you ever see a copy, you should investigate it.

Three References

Then there are three classic texts, all of which I consider to be references, although two of them do have exercises, and hence they are in principle, textbooks. (In fact, both English-language books declare themselves to be both text and reference.)

The first is Kelley. Famed for its exercises (is that a good thing?), and like so many math books of its era, it has no drawings whatsoever. He said he thought of it as “What Every Young Analyst Should Know.” originally published in 1955, it was republished by Springer in 1975 and is still available.

The second is Dugundji. It was allegedly the textbook for my first-year graduate course, but the instructor didn’t use the book at all. To be honest, I don’t know that there even was a “textbook” for general topology back then, other than Kelley – which I bought at the end of the year from another grad student who was bailing on math. When I looked through Dugundji years later, I liked its organization; good enough to make up for its lack of handholding. It’s probably the first place I look to see whether something is true or false. It includes chapters on set theory, ordinals and cardinals. Oh, it has no drawings either.

If you know enough to be looking for reference books in topology, you probably know better than I whether you want to find a copy of Dugundji.

(I am not sure where or when I learned what I know of topology. it might very well have been the combination of Dugundji & Kelley. both are reasonable places to look if you’re tripping over topology in, say, functional analysis. That is, they don’t have a lot of motivation: just the math, ma’am. But when I brought my own motivation, they were fine.)

The third is Seifert & Threlfall, but I list it not because I know it, but for its language. No, not for its exposition, literally for its language. It is available only used, and last time I looked, the cheapest English translation was $300. The book is frequently referenced for its counterexamples, and it is highly praised in books that do more than just list references.

I found a German reprint by Chelsea, in excellent condition, for $35. Having recently read some Harry Potter in German, I think my high school German is good enough for a math book. Since it’s an order of magnitude cheaper, I encourage you to buy the German rather than the English, if you think you can handle the language.

Three Proto-topology

The next group is three books which spend a lot of time on proto-topology, as it were. The first is Kasriel. I love this book because it has, in one volume, and in order, the topology of the real line, of R^n, of metric spaces, and then it does general topology. No running for an advanced calculus book to compare and contrast a metric space theorem with the corresponding R^n theorem. It is typical that I love seeing topology and metric spaces in the context that justifies the generalizations. In contrast to the next two books, you could learn topology out of this book.

Kasriel essentially gives us the historical path to point-set topology.

Unfortunately, Kasriel is out of print. If it sounds interesting, either look for a used copy, or look for another book that has the same organization: lots of R^n, lots of metric spaces, and topology, in one binding.

The second is Naber. This book has a lot of applications of topology, all in R^n, without ever even defining “topological space”! As I like seeing the context of topology in Kasriel, I enjoy the Naber book as a chance to see topology applied. I would not have wanted this book before I knew general topology per se, which defeats its purpose, but I look forward to working thru it someday. It covers simplicial complexes, homology, homotopy, and vector fields, all within R^n.

Be careful when you look for it: the 1980 paperback is out of print, but the book was reprinted in 2000.

The third is an ambitious little book by Chinn & Steenrod. (If you know enough to wonder, yes, that Steenrod.) It’s a lot like the Naber book, trying to do topology without defining it per se. More to the point, this book has two parts: the first wishes to motivate and prove that a continuous real-valued function defined on a closed bounded interval has and attains its extrema, and that y = f(x) can be solved for each y in between those extrema; the second part moves to 2D, where just stating the theorem is an adventure. It has a much simpler goal than Naber, but it’s still a fairly sophisticated goal.

There are things in the second part that I’ve seen but cannot say I’ve understood: specifically, winding number of a curve and index of a vector field. (Did I miss that in Apostol II? Yes! But not by much: it’s only a page or two. I suppose I should be embarrassed but I’ve gotten used to not knowing everything.) Still, I need to work thru this book Real Soon Now. The truly terrifying thing about this book is that its intended audience was “high school students and laymen”. (Was it accessible to high students before the “new math”?)

Be even more careful: although Amazon says it has it in stock, as I write this, it has so many conflicting additional entries saying it’s not available, that I’m a little uncertain.

Two Others

Then, there is a little workbook by Adamson. Too sparse to be a text, but lordy, what a wonderful way to review! 70 pages of definitions and exercises, followed by 78 pages of proofs. You don’t have to hold a piece of paper over the proof to try working it out for yourself. (While checking to see if it’s in stock, I learn that he has also put out a workbook in set theory. hmm….)

Having said that, I must admit that the author intended this as first course, for “independent study based on material carefully prepared by a teacher who otherwise gives minimal assistance.” (If you’ve ever heard of R. L. Moore and his method, you understand.)

Finally, there is a Dover paperback by McCarty. I probably picked it up because it was a Dover, but I might have seen it in a list of references somewhere. It presents just the topology required to get to topological groups, but along the way it has a very nice introduction to commutative diagrams, and the entire book is extremely readable. I sat down to see what was in it, and didn’t put it down until I’d read the whole thing!

The Books Added

Adamson, Iain. A General Topology Workbook. Birkhäuser, 1996.
ISBN 0 8176 3844 X.
[general topology; 10 Nov 2008]
Upper division. From the introduction: “This book has grown from my attempts to provide a self-learning introduction to general topology for several generations of students….” Answers. Guide to further reading.

Chinn, W.G. and Steenrod, First Concepts of Topology. L.W. Singer & Random House1966.
[general topology, algebraic topology; 10 Nov 2008]
From the introduction: “… to show how topology arose, develop a few of its elements, and present some of its simpler applications…. Our presentation … will be centered around two existence theorems… {in one and two dimensions respectively}.” Answers. Guide to further reading (3 books). Epilog.

Dugundji, James. Topology. Allyn & Bacon, 1966.
[general topology; 10 Nov 2008]
Out of print. Reference and text. If they reprint it, i’ll call it a classic. Some homotopy theory.

Kasriel, Robert H. Undergraduate Topology. Krieger, 1977.
ISBN 0 88275 444 0.
[general topology, metric spaces, euclidean spaces; 10 Nov 2008]
Out of print. From R and R^n to metric spaces and then to topology. From the preface: “… it is to {the graduate student’s} advantage to have taken a course in general topology before beginning his graduate program…. essentially self-contained except for elementary calculus.”

Kelley, J. L. General Topology. Springer, 1975 (was van Nostrand, 1955).
[general topology; 10 Nov 2008]
Graduate Text in Mathematics. Classic reference and text. From the preface: “… a systematic exposition of the part of general topology which has proven useful in several branches of mathematics.” Epilog and guide to further reading – in some of the exercises!

McCarty, George. Topology: An Introduction with Application to Topological Groups. Dover, 1988 (orig 1967).
ISBN 0 486 65633 0.
[general topology, topological groups; 10 Nov 2008]
Upper division. How can I not like a book which says, in an exercise, “… make the assumption that every subgroup in sight is closed.” A very readable book. Guide to further reading. Epilog. (Both at end of chapters.)

Munkres, James R. Topology. Prentice Hall, 2000 (2nd ed).
ISBN 0 13 181629 2.
[general topology, algebraic topology; 10 Nov 2008]
A very well written senior and first-year graduate textbook. Lots of words between the theorems provide lots of motivation. If this isn’t the premier textbook, I’d really like to know what is.

Naber, Gregory L. Topological Methods in Euclidean Spaces. Dover, 2000.
ISBN 0 486 41452 3.
[algebraic topology, differential topology, euclidean spaces; 10 Nov 2008]
From the preface: “… to persuade students… that the evolution of topology from analysis and geometry was natural and, indeed, inevitable; that the most fruitful concepts and most interesting problems in the subject are still drawn from independent branches of mathematics…. an ambituous agenda of topics has been included….” Guide to further reading. Answers.

Seifert, H. and Threlfall, W. Lehrbuch der Topologie. Chelsea, 1945.
[general topolgy; 10 Nov 2008]
Out of print. Classic reference. I bought it for the counterexamples; I bought it in German for the price.

Sieradski, Allan J. An Introduction to Topology & Homotopy. PWS-Kent, 1992.
ISBN 0 534 92960 5.
[general topology, algebraic topology; 10 Nov 2008]
Out of print. A very well written senior and first-year graduare textbook. From the preface: “Most topics are developed slowly in their historic manner, in order that a newcomer not be overwhelmed by the ultimate achievements of several generations of mathematicians.”

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.

In contrast, the sphere movie says going thru is ok, but tearing is not. (That’s the same link cited in my original comment.)

Keep it simple. Jim’s comment was that deforming and gluing a flat sheet to form a torus requires another dimension, namely time. To be specific, Jim considered 3 processes:

1. glue the left and right sides together, then glue the two circles.
2. glue the top and bottom sides together, then the two circles.
3. glue both pairs of sides together simultaneously.

He cannot see that the third process generates a torus. Well, neither can I.

I’d like to think that it does, but it doesn’t have to.

The formal mathematics doesn’t actually deal with the deformation process we’re imagining; rather it shows “before” and “after”. Don’t get me wrong: the formal math is motivated by the deformation process, and if the formal math hadn’t led to results which were by-and-large plausible, we would have changed the math.

The mathematics requires that a function describing “before” and “after” be a homeomorphism: 1-1 and both the function and its inverse are continuous. But this isn’t a description of “during”. We describe the result of gluing, not the process of gluing.

Let me show you a much simpler drawing than a torus, simpler even than a cylinder. I start with a line segment of the x-axis, and I want to transform it into a circle. If I imagine that I am wrapping the line onto the circle – wrapping a strip of paper mache’ onto a frame – then I could come up with the following mapping.

It starts at the origin and moves CWW (counter-clockwise); t runs from 0 to .

Now, draw lines from “before” points on the line to “after” points on the circle. Let’s do it first for just the leftmost quarter of the line segment, :

add the next quarter, :

We see a problem as we move toward the top. As one example of many, the line L to the top of the circle, mapping the point on the line to the point (0,2) on the circle, passes through the circle. That is, it passes thru points that have already been mapped to the circle. If we imagine that these lines had been traced out at constant speed, but stopping when they reach their destination on the circle, then the line L hits and goes thru a point on the circle.

The problem is even clearer when we add mappings from the last half of the line segment, since all the lines which are mapped to the left of the circle must pass thru the right of the circle, and all of those points have already been “installed”, as it were. Let’s see this with the 3rd quarter of the line segment :

This process was a far cry from “wrapping” the line segment onto the circle. But such processes have nothing to do with the homeomorphism between the glued line and the circle.

The homeomorphism is a relationship between the glued line and the circle; not a relationship from the glued line to the circle. There are many processes from the glued line to the circle, but not all of them are nice. And this is one that isn’t nice.

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.

So we have the Euler characteristic in one hand. With the other hand, let’s grab the classification theorem. One form said that every compact connected surface is a connected sum of tori T or of projective planes P:

gT, g = 0,1,2,3…

or

gP, g = 1,2,3,…

(where the sphere is 0T, the “torus” with no holes; 2T = T#T, etc., and the integer g is called the genus of the surface.)

Is there any chance we can easily compute the Euler characteristic of the connected sum A#B from the Euler characteristics of A and B? If we could, then we could compute the Euler characteristic of nT and nP, hence of all connected compact surfaces.

Of course we can, and here’s the equation:

Oh, now go get the Euler characteristics of triangulations of the sphere, torus, and projective plane.

Ok, we can do that.

But it would be simpler to get the Euler characteristic from the polygonal disks from which we constructed the basic surfaces. We would have to know that counting the v, e, f (after identifications) of the polygonal disk gives us the same answer, but it does.

Yes, having specifically studied Bloch because he distinguished topological surfaces from simplicial surfaces, I am now effectively going to compute the Euler characteristics of topological surfaces instead of simplicial surfaces.

I never said I didn’t want to mix it all together; I just wanted to see the separate ingredients that went into the stew.

In principal we only need 3 surfaces: sphere, torus, and projective plane. Nevertheless, since the Klein bottle has an equally simple polygonal disk, I’ll show it too. One set of representations is

Edges which are to be identified (“glued”) have the same symbol; an arrow shows the directions which are used. Points which are to be identified have the same color. I considered using colors for the edges, but we can actually do algebra with the symbols: the sphere can be called , the torus , etc.) If it helps, the colors of the vertices are a consequence of the identifications of the edges.

From them, we compute that the Euler characteristics are

Note that the largest Euler characteristic is 2, and it corresponds to a sphere. Note that all four are non-negative. Note that an Euler characteristic of 1 corresponds to the (non-orientable) projective plane. Then for Euler characteristic of 0, we have two surfaces, one orientable, one not.

Then we compute that the Euler characteristic of gT is 2 – 2g, i.e. the set of even numbers less than or equal to 2. And the Euler characteristic of gP is 2 – g, i.e. the set of integers less than or equal to 1. (Maybe I should restrict g > 0 for nP, but I’m not sure it’s worth worrying about.) So if we have an odd negative Euler characteristic, the surface is unique and non-orientable; if the Euler characteristic is even and negative, there are two surfaces, one orientable, one not.

Note that all the new surfaces we construct have Euler characteristic less than zero.

Note that the connected sum of a sphere and any surface A has the same Euler characteristic as A. in fact, the connected sum is homeomorphic to A; gluing spheres to surfaces does not change them.

We know, however, another form of the classification theorem. The non-orientable surfaces gP can be written as a connected sum of tori plus either one projective plane or one Klein bottle:

mT#P

or

mT#K

for some m (and I am paying no attention to the relationships between these m’s and the genus g). To prove that, we need to know that

P # P = K

and

P # P # P = P # T.

We can now compute that the Euler characteristic of mT#P is (2 – 2m) + 1 – 2 = 1 – 2m, i.e. the set of odd integers less than or equal to 1.

The Euler characteristic of mT#K is (2-2m) + 0 – 2 = -2m, i.e. the set of even integers less than or equal to 0.

So, in fact, if the Euler characteristic is an odd negative, there is a unique surface of the form mT#P; if the Euler characteristic is even negative, there is an orientable surface of the form nT and a non-orientable surface of the form mT#K.

References for this material are

• Bloch, “A First Course in Geometric Topology & Differential Geometry”.
• Lee, “Introduction to Topological Manifolds”.
• Firby & Gardiner, “Surface Topology”.
• Massey, “Algebraic Topology: An Introduction” or “A Basic Course in Algbraic Topology”

See the bibliography.

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.