# Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.

The audience is familiar with characteristic classes at the level of Milnor Stasheff. We are no experts on homotopy theory.

What would you recommend?

Edit: Thanks for the answers, I like them! I found it hard to choose one, as most seem like great sources. I chose to accept the lecture notes (even though I was looking for a book) because this seems doing what we want the seminar to be about.

• In my mind Stong's notes on cobordism theory is a nice reference book, not sure whether it is good for seminar. The math language used there is, say, more or less old fashioned. But I think it makes the content easier to understand. – Mingcong Zeng Jan 17 '16 at 8:58
• Great suggestions; you probably have chosen one by now. If I may I wish to add to the great list of suggestions the followings: 1) Buchstaber had a lecture notes on Geometric Cobordism which starts with simple notions and ends which computing with formal groups laws. I suppose you can ask him directly or contact people in Manchester where he gave these lectures. 2) I think Milnor's book Topology from Differential ...'' is still an interesting one. The reader gets the idea how to see (framed-)cobordism as a generalisation of the notion of degree - still a very geometric approach. – user51223 Jan 28 '16 at 22:23

Perhaps the Notes on cobordism by Haynes Miller could be of some help too.

Another possibility (but geared primarily towards applications in symplectic geometry) is the book

V. Guillemin, V. Ginzburg, Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, 98. American Mathematical Society, Providence, RI, 2002.

you ask specifically for a book; one (expensive) option is On Thom Spectra, Orientability, and Cobordism, by Rudyak, announced as

... the first guide on the subject of cobordism since Stong's influential notes of a generation ago. It concentrates on Thom spaces (spectra), orientability theory and (co)bordism theory (including (co)bordism with singularities and, in particular, Morava K-theories). These are all framed by (co)homology theories and spectra. The book is easy to use by students, for when proofs are not given, specific references are.

MathSciNet gives an enthousiastic review, "The book is indispensable for research workers in algebraic topology. The presentation of the material is very nice and thorough, and this makes the book convenient for students with preliminary knowledge of algebraic topology."

• That's a great book, but I don't know how useful it will be for an introductory seminar that doesn't want to delve too deeply into homotopy theory. – Greg Friedman Jan 17 '16 at 19:38

The book Differentiable Periodic Maps by Conner and Floyd is a classic reference. Despite its age, the book is very easy to understand and gives clear expositions of some important topics which are difficult to find elsewhere (such as the bordism spectral sequence and characteristic numbers of maps). As the title suggests, the latter part of the book is about the authors' research on bordism of smooth $$\mathbb{Z}/p$$ actions, which might be a good place to go after you cover the first two chapters.

The only books that I know of have already been mentioned, but Dan Freed has some nice lecture notes (https://www.ma.utexas.edu/users/dafr/M392C-2012/index.html). Little knowledge of homotopy theory is assumed; the concepts are introduced as necessary. The notes touch upon various topics, culminating in an exposition of the cobordism hypothesis.

There is an interesting exposition on RANICKI`s book "Algebraic and Geometric Surgery", chapters 2 and 6. There is a chapter devoted to bundles and it also includes Pontrjagin Cobordism theorem...