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Plans

Posted by Akhil Mathew under

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So MaBloWriMo was kind of a failure for me. I did make the fifteen posts I planned for, but I said much less than I had hoped to, and dropped out near the end of the month. I also did not keep my promise of a series of posts on local cohomology. Those may happen someday, but I do not think I will have the time to do that in the near future.

Nonetheless, it turned out that my commutative algebra course covered most of the material that I was planning to write about (e.g. regular local rings, homological theory, etc.). Since I have taken detailed notes for that class, I still ended up writing about the material in some way. I have created a website where I have posted these notes and many more things.

For the winter, my priority is to study stable homotopy theory. There are other topics that I also want to read about (mostly in algebraic geometry), but I am planning to take a topics course in topology next semester that will require this, so it probably will be the focus on this blog. My knowledge of homotopy theory in general right now is very limited, though. So I hope to begin talking about spectra soon, and with the additional free time that I now have, posts may actually be somewhat frequent.

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December 5, 2010 at 8:17 pm

Have you considered Switzer’s Algebraic Topology-Homotopy and Homology or Bredon’s Topology and Geometry for algebraic topology? Switzer is a excellent book as is Bredon, but perhaps Bredon requires less prior knowledge to read. (In any case, you probably have the prerequisites to read Switzer.) You could also try Bott and Tu’s Differential Forms in Algebraic Topology if you haven’t already seen spectral sequences and characteristic classes. I’d personally recommend going through Bredon and then moving to Bott and Tu, but that depends a lot on how much of Bredon overlaps your current knowledge.

December 5, 2010 at 8:26 pm

Thanks! I have indeed: my current topology teacher had mentioned to me Switzer’s book. I am currently trying to read it. It is admittedly rather dry, and I will need to supplement it with additional sources to get some geometric intuition. I have also seen Bredon’s book and read a fair portion. I have read some of Bott and Tu and plan to study that further along with Hatcher’s _Spectral sequences in algebraic topology_ this winter. I hope to learn about characteristic classes as well. These topics (spectral sequences, characteristic classes, etc) will be covered in the continuation of my current algebraic topology course in the spring (which is separate from the topics course).

Jacob Lurie (who is teaching the topics course) recommended learning stable homotopy theory from a book by Adams on _Stable homotopy and generalized homology._ In addition, I am interested — for the course and for its own sake — in studying the theory of simplicial sets and model categories, for which I’ve been looking at various books: Goerss-Jardine, May, Hovey, and Quillen’s original _Homotopical algebra_ (which is surprisingly readable). The priority for the course was the theory of spectra, but I would be interested in learning these various topics. I’m not sure which of them will turn into blog posts.

December 6, 2010 at 10:26 pm

Sounds good. Jacob probably knows you better. If you’ve read Spanier, then you’ll probably find that the style is similar to Switzer because both are very abstract (Spanier is a bit more basic though). I think you’ll also find that as you go further in algebraic topology, the geometric flavor of the subject tends to be less represented in modern texts. Hatcher is excellent for geometric flavor, but you’re probably already familiar with the material there. (Hatcher is an excellent book but it seems to be one of those books that people either “like” or “don’t like”.) If you haven’t read through Hatcher, it might be good to see things you’re already familiar with there and how they are geometrically motivated. Then, when you read more modern texts, you can make it an exercise to try to see how to geometrically motivate the homological algebraic proofs for yourself. Bott and Tu is very good for spectral sequences, and the geometry is certainly there, so if you’re reading that, that’s great too. Another book that’s slightly less advanced, but introduces some of the more basic aspects of homotopy theory, is Brayton Gray’s Homotopy Theory: An Introduction to Algebraic Topology. Perhaps a better level for you would be Hu’s Homotopy theory which is also good. I probably shouldn’t suggest this, but Whitehead’s Elements of Homotopy Theoy is quite a comprehensive book if you’re willing to put in (at least a year of) hard work. But I expect that to be usually recommended to graduate students (and even algebraic topologists) who want to specialize in algebraic topology. You might like it but it seems to be one of those books to come back to rather than to read for a first time. Finally, you might like to try Munkres’ Elements of Algebraic Topology. It might be a bit basic but Munkres is really nice to read, and it’s good for the other side of algebraic topology: homology and cohomology, if you want to learn more about that. But I’d certainly encourage you to complete Bredon because the material is Bredon is a good chunk of what every topologist should know. If you’ve gone through it, then that’s very good.

December 6, 2010 at 11:51 pm

Yes, it usually helps me to see a bit of geometry (as in Hatcher) when I learn something for the first time. I appreciate the fact that Spanier and Switzer tend to be much heavier on category theory than Hatcher does, though.

Lurie actually mentioned Whitehead’s rather comprehensive book to me: I was not successful in making much progress on it (though I learned a few things about cofibrations and fibrations). One of the problems that I had at encountering on first glance was that it has the same Bourbakian style but without the cleaner modern machinery (Whitehead says he makes maximal use of elementary methods)—for instance, I find it nicer to prove the Freudenthal suspension theorem (or more generally, homotopy excision) using a spectral sequence. On the other hand, I still don’t fully grok the Serre spectral sequence (I understand that it comes directly from a double complex, but I haven’t fully worked through the algebra that the page is what it is).

I remember looking at Munkres’s book last summer, and it helped me understand what acyclic models were about. It is, indeed, fun to read. I have not tried Hu’s book; I’ll take a look.

December 7, 2010 at 8:38 pm

I think if you read a bit further into Bott and Tu, you’ll see spectral sequences explained with a geometric flavor.

Whitehead is a good textbook but the problem is that it’s too big to be used as a textbook. It might be better for self-study, however. It depends very much on your efficiency, or how long you think it will take you to read Whitehead. If it is going to take a year or more, then I would generally recommend otherwise since one year is a long time, and you might want to concentrate on a broad knowledge of algebraic topology, rather than an in-depth knowledge, since you still (I’m guessing) don’t know whether you’ll end up researching it. For that reason, Hu’s book is ideal: it’s reasonably sized and can be completed in 6 months. This all depends on your efficiency, however. I’ve known fairly strong graduate students who would take 2 years to read Whitehead. (But these students would generally spend a lot of time thinking about the material and that might be the case with you too.)

So it might depend on how you actually read the text. If you’re the sort of person who likes to read books linearly, go through the proofs, try to pick up the main ideas, do the exercises etc., then it might take you a long time. On the other hand, if you’re the sort of person who likes to skip around the text and get a feel for the material rather than trying to completely digest each and every proof, then it might take you quicker. That said, both ways may be quick for you. Judging from your blog, you seem to be the sort of person who can pick up mathematics quickly and that’s an important skill.

But returning to the point, I think you’ll quite like Hu if you liked Munkres. To someone who wanted to get a broad overview of algebraic topology, I’d recommend them to read Elements of Algebraic Topology, Brayton Grays Homotopy Theory: An Introduction to Algebraic Topology, Hu’s Homotopy Theory, and then Bott and Tu’s Differential Forms in Algebraic Topology. Of course, the student would also want to supplement their algebraic topology with some smooth manifolds as well; Bott and Tu does that a bit, but Do Carmo is good for that too.

December 7, 2010 at 9:49 pm

Yes, that does seem to a problem with Whitehead. We used Hatcher in my algebraic topology class (introductory, so at a lower level). On the other hand, a large size tends to pique my curiosity, and from what you said I have a sudden urge to try to read it cover to cover :-). I don’t know whether I want to study algebraic topology, though I think I would like to study something between that, algebraic geometry, and commutative algebra (you mentioned earlier that you advised getting up to research level in three fields by graduate school — I think I will aim for those three). All the three tend to have a primarily algebraic flavor to them. I understand that quite a few people tend to straddle these three areas, presumably because of their connections (I suppose Lurie is one of the most famous).

The way I read tends to depend on the type of the book. What I enjoy most are books where it *is* possible to read mostly linearly (e.g. EGA) and where things are explained in significant detail, while occasionally skipping irrelevant details. If that can be done for Whitehead, then perhaps I will enjoy

it. I suspect I may have more success with it now than I did earlier, as I have a bit more experience with this material.

Hu’s book does indeed look nice. I’ve added that to my winter reading list. It looks like another source for spectral sequences, in fact! This is precisely what I would like to learn (excluding the simplicial stuff, for which I have other sources anyway).