Algebraic topology beyond the basics: any texts bridging the gap?

Question

Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts - both geometric ones like Allen Hatcher's and algebraic-focused ones like the one by Rotman and more recently, the beautiful text by tom Dieck (which I'll be reviewing for MAA Online in 2 weeks, watch out for that!) - there are almost no texts which bring the reader even close to the frontiers of the subject.

GEOMETRIC topology has quite a few books that present its modern essentials to graduate student readers - the books by Thurston, Kirby and Vassiliev come to mind - but the vast majority of algebraic topology texts are mired in material that was old when Ronald Reagan was President of The United States. This is partly due to the youth of the subject, but I think it's more due to the sheer vastness of the subject now. Writing a cutting edge algebraic topology textbook - TEXTBOOK, not MONOGRAPH - is a little like trying to write one on algebra or analysis. The fields are so gigantic and growing, the task seems insurmountable.

There are only 2 "standard" advanced textbooks in algebraic topology and both of them are over 30 years old now: Robert Switzer's Algebraic Topology: Homology And Homotopy and George Whitehead's Elements of Homotopy Theory. Homotopy theory in particular has undergone a complete transformation and explosive expansion since Whitehead wrote his book. (That being said, the fact this classic is out of print is a crime.) There is a recent beautiful textbook that's a very good addition to the literature, Davis and Kirk's Lectures in Algebraic Topology - but most of the material in that book is pre-1980 and focuses on the geometric aspects of the subject.

We need a book that surveys the subject as it currently stands and prepares advanced students for the research literature and specialized monographs as well as makes the subject accessible to the nonexpert mathematician who wants to learn the state of the art but not drown in it. The man most qualified to write that text is the man to uttered the words I began this post with. His beautiful concise course is a classic for good reason; we so rarely have an expert give us his "take" on a field. It's too difficult for a first course, even for the best students, but it's "must have" supplementary reading. I wish Dr. May - perhaps when he retires - will find the time to write a truly comprehensive text on the subject he has had such a profound effect on. Anyone have any news on this front of future advanced texts in topology?

I'll close this box and throw it open to the floor by sharing what may be the first such textbook available as a massive set of online notes. I just discovered it tonight; it's by Garth Warner of The University Of Washington and available free for download at his website. I don't know if it's the answer, but it sure looks like a huge step in the right direction. Enjoy. And please comment here. http://www.math.washington.edu/~warner/TTHT_Warner.pdf

Answer 1

At the moment I'm reading the book Introduction to homotopy theory by Paul Selick. It is quite short but covers topics like spectral sequences, Hopf algebras and spectra. This is the first place I've found explanations (that I understand) of things like Mayer-Vietoris sequences of homotopy groups, homotopy pushout and pullback squares etc.. The author writes in the preface that the book is inteded to bridge the gap which the OP talks about.

Answer 2

Good lord, Charles, was the reposting of this an invitation for another advertisement from me? ``More concise algebraic topology. Localization, completion, and model categories'', by Kate Ponto and myself, is available for purchase and will be formally and officially published next month. I have a copy in my hand, and the final version is 514 pages (including Bibliography and Index). Still 65 dollars (and don't fall for pirate editions on the web). It is not perfect, of course. (I know of one careless mistake every reader will catch and one subtle mistake almost no reader will catch). I offer $10 to any reader finding a mistake I don't know about, even misprints. The book is intended to help fill the gap (and another, more calculational, follow up to Concise is planned). The first half covers localization and completion and is more technical than I hoped simply because so much detail was needed to fill out the theory as it was left in the great sources from the early 1970's (Bousfield-Kan, Sullivan, Hilton-Mislin-Roitberg, etc), especially about fracture theorems. The second half is an introduction to model category theory, and it has a number of idiosyncratic features, such as emphasis on the trichotomy of Quillen, Hurewicz, and mixed model structures on spaces and chain complexes. The order is deliberate: novices should see a worked example of serious homotopical algebra before starting on categorical homotopy theory. There is a bonus track on Hopf algebras for algebraic topologists and a brief primer on spectral sequences. There are example applications sprinkled around, although more might have been desirable. The book is quite long enough as it is. Merry Christmas all.

Answer 3

I'm absolutely thrilled by the existence of Strom's "Modern Classical Homotopy Theory":

http://www.ams.org/bookstore-getitem/item=GSM-127

Makes for essential reading, I think. Warmly recommended.

Answer 4

Homotopic Topology by Fuchs, Fomenko, and Gutenmacher, mentioned above by Ilya Grigoriev, is a wonderful book which is practically unknown here (english version was done by an obscure eastern european publisher and has been out of print for decades) and hard to get even via an interlibrary loan. It's now availaible in pdf at

http://www.math.columbia.edu/~khovanov/algtop2013/

although the files are pretty large.

Answer 5

Aside from "textbooks", there are quite a few more informally prepared lecture notes. Many of these are available online, but often aren't well advertised or easy to find, since they usually don't get published or make it to arxiv. I'll list a couple I know about (I attended some of these courses), and I'll wiki this answer so people can add more. The kinds of course notes I have in mind are ones that introduce or cover some big modern topic, rather than ones which are geared to proving one theorem.

• Haynes Miller's course on Cobordism, (notes by Dan Christensen and Gerd Laures). An introduction to the Steenrod algebra, cobordism, formal groups, $MU$ and $BP$, and much more.

• Haynes Miller, course on homotopy theory of the vector field problem, part 1 and part 2, (handwritten notes my Matt Ando). Covers classical topics related to the vector field problem, the EHP sequence, and Adams's work on Im(J). These are available in incomplete form in a TeX document.

• A couple of notes from courses by Mike Hopkins on elliptic cohomology and related stuff: 1995, 1999.

• Jacob Lurie is currently teaching a course at Harvard about chromatic homotopy. He's posting his lecture notes, and Chris Schommer-Pries is also posting notes.

Anything else?

Answer 6

The "word on the street" is that Peter May in collaboration with Kate Ponto is writing a sequel to his concise course (with a title like "More concise algebraic topology"). I've seen portions of it, and it seems like it contains nice treatments of localizations and completions of spaces, model category theory, and the theory of hopf algebras. I have no idea what else it might contain or when it will be released, but if you are interested it might be worth writing to either of the authors for more details.

Answer 7

The standard texts (Hatcher, May, etc.) cover material which was, in large part, understood by 1950, though this material is filtered - conspicuously so, in May's text - through the authors' modern perspectives and sensibilities. That leaves another half-century of development. So, as a follow-up to first-year algebraic topology - still far from the cutting edge, but very relevant to reaching it - may I recommend reading some of the classic papers of the mid-twentieth century?

Many are easy to find online. Several - no, many! - are written by great mathematicians and great expositors. For me, the material is the more exciting in the words of its discoverers. Many people will have their own favourites; my list is slanted towards differential topology.

A couple by Serre. Homologie singulière des éspaces fibrés has as clear and economical an account of spectral sequences as I've seen anywhere. The method of Groupes d'homotopie et classes de groupes abéliens might be considered old-fashioned, but it gives a strong taste of what localisation can achieve (e.g. where is the first $p$-torsion in the stable stems?).

Three papers that achieve perfect marriages of algebraic topology and differential geometry: Thom's Quelques propriétés des variétés différentiables founded cobordism theory. Kervaire-Milnor's Groups of homotopy spheres I essentially began surgery theory. Both are astonishingly far-seeing. Finally, Deligne-Griffiths-Morgan-Sullivan's Real homotopy theory of Kähler manifolds: minimal models are things you can build at home!

Answer 8

I have written up some detailed lecture notes

Introduction to Stable homotopy theory

Prelude -- Classical homotopy theory (pdf, 99 pages)

Part 1 -- Stable homotopy theory

• Part 1.1 -- Sequential spectra (pdf, 67 pages)

• Part 1.2 -- Structured spectra (pdf, 80 pages)

Part 2 -- Adams spectral sequences (pdf, 56 pages)

This introduces and then proceeds systematically via model categories. Full details and proofs are given.

To view the web version you need the Firefox browser

Answer 9

UPDATED. After Peter May and Kate Ponto released their new book, there are very readable introductions to many of the topics on the "second level" of algebraic topology.

• There is a wonderful book on Cohomology Operations by Mosher and Tangora. It is thin (and only discusses one topic), but very nice.

• May & Ponto's new book is very nice. It covers three topics (Professor May's comment above has details) + an appendix on spectral sequences, which is short but very much to the point. I used to fear that any book by May was secretly about category theory, but that is not true about 3/4 of this one (unless the secret is hidden too well).

• There is a pretty good, and comprehensive book by Fomenko and Fuks (or Fuchs?) on homotopy theory. I've only seen the Russian version (so I can't vouch for the translation). It's also not very well-known, and not very easy to find, which is a shame (the Russian version is more obtainable). It has a lot of stuff, including one of the nicer introductions to spectral sequences (although I don't know a single book that does this well. Serre's thesis is nice, Hatcher's notes are OK, but this seems to be a topic best learned in a good class). It's also very readable. Here's a review (institutional access probably required) with a description of its contents.

Answer 10

First of all I want to comment that beyond the two "standard textbooks" Andrew L. mentioned (Switzer and Whitehead), there is at least Adams's classic "Stable Homotopy and Generalised Homology", which goes in some aspects deeper than Switzer. Besides this, I want to mention two more recent books:

• Neisendorfer's Algebraic Methods in Unstable Homotopy Theory. It is perhaps not perfectly edited, but seems to essential reading as a source for modern unstable homotopy theory.
• Kochman's Bordism, Stable Homotopy and Adams Spectral Sequences. It deals pretty early with spectral sequences and proves some standard results in classical homotopy theory via them (like Blakers-Massey, Freudenthal..) In then goes on to introduce the stable category and compute the first 20 stable homotopy groups of the sphere (or so). It also treats characteristic classes and bordism. The treatment of the stable category is not really to my taste, but the book definitely deserves a look.

Answer 11

Perhaps the book of Douglas Ravenel, http://www.math.rochester.edu/u/faculty/doug/preprints.html

Answer 12

I learned -still and will be learning - the fundamentals of Algebraic topology from a professor at my University, Dr. Carlos Prieto. He is the co-author, along Dr. Samuel Gliter and Dr. Marcelo Aguilar, of a great book that covers the fundamentals of Algebraic Topology, they also treat K-Theory, Vector Bundles &, one of the great things, they give and alternative and interesting proof of the Bott periodicity theorem.

The book I am referring to is the following:

http://www.amazon.com/Algebraic-Topology-Homotopical-Viewpoint-Universitext/dp/1441930051

They are currently working on the second edition of the book.

This is the web page of Dr. Carlos Prieto:

http://www.matem.unam.mx/cprieto/