What Are Your Recommendations for Algebraic Topology Textbooks?

[R.P.F.]

Hey guys,

I want to study algebraic topology on my own. I just finished a semester of pointset topology and three weeks of algebraic topology. We did not use a textbook. Can anyone recommend a book on algebraic topology?

Hatcher is fine but it is not as rigorous as I want. Munkres has a book on algebraic topology but it is kind of out of date. If no one has a better suggestion then I will probably switch between Hatcher and May. But I want to try my luck here first. Thanks!

 

[mathwonk]

I am puzzled by your criticism of Hatcher. Maybe I have not read it carefully but it seemed rigorous. Do you mean not clear or precise enough? The professional topologists in my department swore by Hatcher.

Have you looked at Spanier? It seems extremely rigorous. What about Dold? The usual problem with algebraic topology books is they are hard to understand, not that they are unrigorous.

As for books that I liked for their efforts to explain well, I always liked anything by Massey. William Fulton also writes well. I also really liked the little book by Artin and Braun, but it is much more elementary than May. The popular book by Marvin Greenberg is essentially an expansion of the book by Artin and Braun. Vick is also very clear. I always found Munkres a little tedious myself, but some of our professional geometers liked him a lot.

May of course is very authoritative but to me, even as a professional mathematician, I think it is very terse. One book that is maybe not as rigorous as you like, but is very well written for comprehension, is that of Bott and Tu, written by the superbly skilled Loring Tu, probably from lectures of Bott. That would probably be my own favorite.

But I am not quite sure what you want if May looks more appealing to you than Hatcher. You might like May better.

There is an interesting selection of older but original readings mostly from research journals by J. Frank Adams you might enjoy: Algebraic topology; a student's guide.

http://www.abebooks.com/servlet/Sea...=algebraic+topology+student's+guide&x=58&y=14

 

[deluks917]

What do people think of Topology and Groupoids?

https://www.amazon.com/Topology-Groupoids-Ronald-Brown/dp/1419627228

 

[jgens]

If no one has a better suggestion then I will probably switch between Hatcher and May. But I want to try my luck here first. Thanks!

I actually think a combination of Hatcher and May works very well. May is very dense, but once you have some understanding of the material, I think the way he presents things is actually pretty illuminating. The categorical perspective that May develops is also handy. For example, once you have read through May's section on the van Kampen Theorem, go read the section on the van Kampen Theorem in Hatcher's book and recast all of the arguments in categorical terms. Often times you can clear up arguments that don't seem particularly rigorous using May's version of the van Kampen Theorem.

What do people think of Topology and Groupoids?

I have only skimmed Topology and Groupoids, but from what I can remember, I do not think that it is more rigorous than Hatcher. On top of that, it omits important topics like homology and cohomology. On the other hand, the text focuses a lot on the fundamental groupoid of a space, and this is an interesting approach to things.

So, I would say Brown's book probably is not the best primary source to learn algebraic topology from. However, I think it is definitely a source worth reading to learn about some interesting topics that are omitted from most standard texts on the subject.

 

[R.P.F.]

I am puzzled by your criticism of Hatcher. Maybe I have not read it carefully but it seemed rigorous. Do you mean not clear or precise enough? The professional topologists in my department swore by Hatcher.

Hi mathwonk,

I guess I was not criticizing Hatcher. Hatcher does a great job of explaining the ideas. But sometimes the notations it uses do not make much sense.(I have only read the chapter on fundamental groups and I found that problem.) I think it can be made more rigorous. I am aware that May is dense and that is why I have to read it together with Hatcher.

Thanks for your suggestions! I will definitely look up the books you mentioned. I got all winter break to do this.

 

[R.P.F.]

I actually think a combination of Hatcher and May works very well. May is very dense, but once you have some understanding of the material, I think the way he presents things is actually pretty illuminating. The categorical perspective that May develops is also handy. For example, once you have read through May's section on the van Kampen Theorem, go read the section on the van Kampen Theorem in Hatcher's book and recast all of the arguments in categorical terms. Often times you can clear up arguments that don't seem particularly rigorous using May's version of the van Kampen Theorem.

I also really like the way that May presents the material. Thanks!

What do people think of Topology and Groupoids?

https://www.amazon.com/Topology-Groupoids-Ronald-Brown/dp/1419627228

I know several people who read/skimmed it and they all said that it's pretty 'unusual'. As jgens said, if you want something extra, it might be a good idea.