Algebraic topology. A students guide
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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.
ISBN 13: 9780521080767
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.
Algebraic topology : a student's guide
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.
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Skip to navigation. Algebraic Topology is rich and diverse area of mathematics that develops and applies a variety of algebraic, categoric and combinatorial techniques to solve problems in many areas of maths.
It was originally born in the 's and 30's to study various topological and geometric issues, but since then has had spectacular effect in areas such as number theory, mathematical physics, algebraic geometry, dynamics, logic, algebra, computer science and even social theory, and it continues to provide foundational insight into core areas of geometry and topology. In Leicester we have a very active research group that covers both fundamental work within algebraic topology and homological algebra as well as some of its important applications to other parts of mathematics.
Current research of members of the group includes both stable and unstable homotopy theory and their relation to algebraic geometry, category theory, dynamical systems, mathematical physics, structured spectra, homological algebra, aperiodic tilings and moduli spaces and stacks of various kinds. Leicester is home to the London Maths Society funded "Transpennine Topology Triangle" TTT , a joint topology seminar run with the Universities of Manchester and Sheffield at which staff and students meet about 6 times a year for talks, collaboration and generally keeping in touch with the topological community in England.
Leicester has also regularly hosted the British Topology Meeting as well as workshops on the interfaces with other areas of mathematics such as "Algebraic methods in geometry and physics" in and "Aperiodic Order" in Current members. Undergraduate: mathsug le. Student complaints procedure.
MATH5665 Algebraic Topology
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