Thursday, January 20, 2005

Betti Numbers

It's been many days now since Afra Zamorodian announced his new monograph 'Topology and Computing'. Now that I finally got a chance to read an excerpt, I am encouraged ! Finally, I am hoping that there is a way to understand Betti numbers without needing an entire year's worth of study :).

Great job, Afra ! Topology is an area that has many intriguing connections with geometry and algorithms in general (Kneser's conjecture, the Kahn-Saks-Sturtevant partial resolution of the evasiveness conjecture), and we need books that can bridge the gap. Another "geometric' text is the wonderful book by Matousek on the Borsuk-Ulam theorem, and a more 'combinatoric' text in this regard is the survey 'Topological Methods' by Björner in the Handbook of Combinatorics.

1. What's so complicated about Betti numbers?

Posted by AC

2. well for starters, the fact that you need to go thru half a book in combinatorial topology to acquire enough machinery to understand their definition. I have heard enough of "first Betti number is number of connected components, and second Betti number is number of holes..er..tunnels.. something...". I'd like something more intuitive but less rigorous so I can approach the formal definition more easily

Posted by Suresh

3. I suppose I did the "going through topology for months" when I was young enough not to feel it much. You should go straight for a good book on Algebraic (not Combinatorial) Topology, is what I say. Then again I don't know if I'd recommend this to someone at your age (ahem!).

Posted by AC

4. yeah yeah, rub it in :). really though, I prefer to read a book with some intuition in the back of my head. it helps to make the constructions a lot easier. I am working thru some books in algebraic/combinatorial topology right now. there is a nice book by Maunders(?) from Dover that I have been plowing thru.

Posted by Suresh

5. Im seeking a good understandable definition of betti number and have not found one yet. It may be too abstruse to define in words. I have three
topology texts and have just ordered Intro to Topology by Bert Mendelson...perhaps there is a good defn in there but I doubt it.

Posted by jurgen

6. Good luck finding one :). I really doubt that Betti numbers are amenable to easy explanations, although after reading Penrose's new book, anything is possible.

Posted by Suresh

7. Whats the title of the new book? I never did read the Emperors New Mind. I did read the brief Penrose / Hawking book, which just posed a lot of questions.

Posted by jurgen

8. It's called 'The Road to Reality'. I highly recommend it. I will probably post a brief review once I read it (I have it ordered), but what little I saw of it in the bookstore convinced me.

Unlike ENM, this is apurely technical book that attempts to survey the vast tracts of mathematics and physics needed to understand modern theories of physics. Any book that can make covariant derivatives make sense deserves to be bought !

Posted by Suresh

9. That intro from Topology and Computing is interesting and clear. Let me know if the rest of it is as lucid. A little expensive for me, I get all my technical books from Dover publications.

Posted by jurgen

10. Check out Afra's web page at Stanford. His thesis is online and the first few chapters provide a good intro to the area.

Posted by Suresh

11. Afra writes clearly and is 'cool' too.. reminds me of Richard Feynman. Excellent color graphics. I downloaded the entire thesis !
I noticed in figure 2.6 of Topology and Computing it says radiuses. Shouldn't it be spelled radii?

Posted by jurgen

12. it is simple maybe in definition but I really don't know how to calculate them for an arbitrary manifold?Do you know a book or a monograph that shows calculating them by algebraic methods?