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 \chi = v - e\ . We can show, for example, that every tree (a graph with no closed paths) has \chi = 1\ . 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.
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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.


The Euler characteristic \chi 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

\chi = v - e + f\ ,

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.
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Topology: separation axioms


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


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
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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)


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

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Books added: no, but additional information added

I have made a major edit to the bibliography. I have tried to mark 3 pieces of information for each book. To be specific, I have tried to mark affirmative answers for 3 questions. (A search should find only affirmative answers.)

Answers. Does it have answers or hints for “many” problems? If so, that is one more reason why the book may be suitable for self-study without a teacher.

Guide to further reading. Not just a bibliography; not just detailed references; but “if you want to know more about this, consider these books, and for more about that, look at those books….” This was prompted by the magnificent guide in Bloch. I wish it were always so clear-cut. O’Neill’s “Elementary Differential Geometry” (2nd ed.), for example, has a short bibliography (8 books), which leads me to say, “I could afford to own all these.” (I own all but one; maybe I should rectify that just for completeness. I like short bibliographies.) Furthermore, he has a sentence about 2 of them. Technically, he has a guide to further reading, and I said so; but I qualified it as only two books. OTOH, a few books break things down into subcategories, but unless they said something about individual books, I didn’t view that as a guide. (The same effect is achieved by end-of-chapter bibliographies, and it’s not what I’m trying to flag.)

Epilog. What might we find in the next book? Not appendices, but for example, a final chapter entitled “advanced topics” – specifically because he said he couldn’t cover them but he wanted to mention them. This was prompted by a marvelous epilog in Massey’s “Algebraic Topology: an Introduction”.

I have tried to set it so that searching the bibliographies page for “guide”, “answers”, or “epilog” will find no extraneous occurrences of these words, except for titles which contain those words.

Finally, the edit date for every book affected by any of these changes is 5 Nov 2008, so you can search the bibliography for 5 Nov to find these changes. That is, until I have some reason to make another edit to any one of the books affected today.

No new books were added today.

I tried to be careful, both looking thru books and editing the bibliography, but don’t bet the farm on my tags; and remember that some of my editions are not current.