# Algebraic Topology Beyond the Basics:Any Texts Bridging The Gap?

Peter May said famously that algebraic topology is a subject poorly served by it's textbooks. Sadly,I have to agree. Although we have a frieghtcar full of excellent first-year algebraic topology texts-both geometric ones like Allen Hatcher's and algebraic-focused ones like 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 it's modern essentials to graduate student readers-the books by Thurston, Kirby and Vasselev 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 its 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 Amd 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 mathematicain 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 true 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 Washingon 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

-
There appears to be a small ratio of question to soapbox. While I think I admire your zeal, it might be wise to take more pauses for breath, or indeed for new paragraphs. –  Yemon Choi Mar 13 '10 at 7:42
I've inserted some paragraph breaks just so that I could read you question - but really, it's a rant, not a question. If I pretend you are asking a question, I think it must be: "What are good modern textbooks on algebraic topology/homotopy theory? If such a thing doesn't exist, what should a good modern textbook contain?" –  Charles Rezk Mar 13 '10 at 16:37
Uh,I don't have one,sorry,fpqc.I DO have a Facebook page and blog,though-the blog is "Tables,Chairs and Beermugs" and it can be found easily through Google.Don't have a link handy at the moment. –  Andrew L Mar 13 '10 at 23:14
I don't understand why the comment above mine (two above this current comment) was voted up, as it doesn't make any sense at all in context. –  Harry Gindi Mar 14 '10 at 1:34
I used to think that people did pointless things on purpose just to annoy me. Now I know that they do. =p –  Harry Gindi Mar 14 '10 at 13:13

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.

-
I have Selick-it just arrived,in fact. I was quite disappointed given the price,but it does contain a lot of nice stuff not accessible in the usual textbook fare. I just think it's ridiculous to pay 50 bucks for what's basically a glorified set of lecture notes. The Fields Institute has a lot of nerve,really...... –  Andrew L Mar 14 '10 at 5:14

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. - To avoid a misconception: The question was not posted by Charles (Rezk), but by Andrew L. Charles Rezk only did some reformatting/writing which affected large parts of the post so that he got assigned the largest share of the CW post and thus shows up prominently. For detailed information on who wrote what, click the date in the middle of the question. – quid Dec 27 '11 at 0:06 Thanks, I'm still a novice at understanding how these things work. Incidentally, in view of the original question, I have no intention of retiring any time soon. – Peter May Dec 27 '11 at 0:24 That last part is VERY good news indeed,Professor May. (I know it's proper in scientific circles to refer to professors by their first names in informal conversation.But I don't feel that's appropriate without asking your permission first.So I hope you don't mind the honorific reference.) I'm a big fan of your work in general and I'm very much looking forward to the finished book.I'm also hoping we can meet and talk shop eventually before you retire-hopefully before I get my PHD. Happy Holidays to you as well! – Andrew L Dec 28 '11 at 21:00 add comment 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. - That's very encouraging,I may do that this summer. May is one of the Gods of the subject and anyone interested in topology owes it to themselves to read his papers and books. – Andrew L Mar 13 '10 at 20:28 Based on first-hand comments from Peter a few days ago, the text is now done, with precisely the table of contents you describe, and will appear soon. – L Spice Mar 13 '10 at 20:31 @Prof. Spice, is there anything like an ETA on it? – Harry Gindi Mar 14 '10 at 1:06 @Harry amazon.com/More-Concise-Algebraic-Topology-Localization/dp/… January? And it's 65 DOLLARS FOR A LITTLE BLUE BOOK?!? Sigh. Rank hath it's priviledges............. – Andrew L Aug 11 '11 at 6:04 @Andrew L : I'd hardly call$65 for a 384 page hardcover book absurd. It's pretty much average or below average for scientific books. Trust me, neither of the authors will get rich on it... –  Andy Putman Aug 12 '11 at 3:43
show 1 more comment

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.

-
This book's getting a lot of press and word of mouth buzz lately-apparently it's been Strom's baby for almost a decade now. The finished book apparently bears little resemblance to the draft lecture note versions that have been circulating the internet for a few years now. In any event,it's very daring to write an algebraic topology problem course,especially one that's supposed to be somewhat more advanced then the usual books.But from what I've seen and heard,this could become one of the gold standard texts for a long time. –  Andrew L Jan 20 '12 at 9:13

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.

-
Welcome to Math Overflow, Mikhail, and thank you for putting the English version of this wonderful book on the Web! –  Mikhail Borovoi Feb 9 '13 at 17:46
The pictures alone make this book worth a look, thank you for this. –  Vidit Nanda Feb 9 '13 at 20:39
@Mikhal Thank you so much for your wonderful addition to the online sources available to students on advanced topics! I plan to spread the word at my coming website bibligraphy,which I hope will be online by next month. –  Andrew L Feb 13 '13 at 0:15

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). Somebody should TeX these ...

• 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?

-
Thanks so much for the wonderful references-these notes aren't really advertised,as you know.The Lurie course I was aware of,the others not so much. All helpful,thanks.I'll have to let you know about a couple of things I'm in the know about when I have time. –  Andrew L Mar 13 '10 at 19:44
Can I suggest that some such site as the n-Lab be asked it a list of such sources be built up so that people can find the links quickly? The readership of the n-Lab is not restricted to Algebraic Topologists of course, but the above list of notes would be useful to have readily available. –  Tim Porter Mar 14 '10 at 7:19

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!

-
Thank you Professor Perutz!I don't think you're still at Columbia,are you,by the way? –  Andrew L Mar 13 '10 at 21:21
The Kan seminar (18.915) at MIT does exactly this every fall -- see www-math.mit.edu/~mbehrens/18.915/index.html for two lists of suggested papers. –  Steven Sivek Mar 13 '10 at 21:30
@Andrew - no, I'm now at UT Austin. –  Tim Perutz Mar 13 '10 at 21:45
Frank Adams's book "Algebraic topology: a student's guide" is a collection of such classic papers. –  Robin Chapman Mar 13 '10 at 21:58
I hope to be taking Dennis Sullivan's advanced topology course and string theory seminar at The City University Of New York Graduate Center next year while my PHD program applications are processed.I'm hoping that will not only expose me to the cutting edge,but allow me to work with one of the greats. –  Andrew L Mar 13 '10 at 22:22

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.

-
I'm aware of the book you mean,Ilya.It's nearly impossible to find anywhere now and if a copy is available,I'm sure it'll cost a king's ransom. I'm a HUGE fan of Russian mathematics texts despite not knowing Russian-fortunately,there are many excellent translations available.In fact,my overall favorite introductory topology texts are the terrific 2 recent ones by V.V.Pravolov avaliable through the AMS. –  Andrew L Mar 13 '10 at 22:25
@Andrew: I'm hoping college libraries will have it. –  Ilya Grigoriev Mar 14 '10 at 3:42
Since there are many answers I provide the link to another one relevant to the last mentioned book mathoverflow.net/questions/18041/… –  quid Feb 9 '13 at 15:10

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.
-

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

-
There are two books by Ravenel - the green and the orange one as is used to call them. Both have a number of misprints, but are essential reading if one wants to get into modern stable homotopy theory. –  Lennart Meier Nov 12 '12 at 11:27

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/

-