It seems that you're in Germany. We have a dedicated site for Germany. This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows:. The underlying theme of the entire book is the Eckmann-Hilton duality theory.

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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!

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. 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 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.

And please comment here. 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. Good lord, Charles, was the reposting of this an invitation for another advertisement from me? 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 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.

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 '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. 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. 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.

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, course on homotopy theory of the vector field problem, part 1 and part 2 , handwritten notes my Matt Ando. 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: , 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. The standard texts Hatcher, May, etc. 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! 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. Kervaire-Milnor's Groups of homotopy spheres I essentially began surgery theory. Both are astonishingly far-seeing. 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. There is a pretty good, and comprehensive book by Fomenko and Fuks or Fuchs? 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. Introduction to Stable homotopy theory.

Prelude -- Classical homotopy theory pdf , 99 pages. Part 1 -- Stable homotopy theory. Part 1. Part 2 -- Adams spectral sequences pdf , 56 pages. This introduces and then proceeds systematically via model categories. Full details and proofs are given. First of all I want to comment that beyond the two "standard textbooks" Andrew L.

Besides this, I want to mention two more recent books:. 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. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 10 years, 2 months ago. Active 4 months ago. Viewed 21k times. While I think I admire your zeal, it might be wise to take more pauses for breath, or indeed for new paragraphs. If such a thing doesn't exist, what should a good modern textbook contain?

Don't have a link handy at the moment.


Introduction to Homotopy Theory

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree.


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Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. This text is based on a one-semester graduate course taught by the author at The Fields Institute in fall as part of the homotopy theory program which constituted the Institute's major program that year. The intent of the course was to bring graduate students who had completed a first course in algebraic topology to the point where they could understand research lectures in homotopy theory and to prepare them for the other, more specialized graduate courses being held in conjunction with the program. The notes are divided into two parts: prerequisites and the course proper. Part I, the prerequisites, contains a review of material often taught in a first course in algebraic topology.

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