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Complex cobordism and formal groups. (English. Russian original) Zbl 1275.55004
Russ. Math. Surv. 67, No. 5, 891-950 (2012); translation from Usp. Mat. Nauk. 67, No. 5, 111-174 (2012).
This is a survey article covering aspects of the mathematics and history of a topic within Algebraic Topology that has assumed great importance for the past forty years or more. It is written by one of the major contributors to its development from the 1960s and onwards, and is very informative on some of the key ideas. It is also filled with comments on minutiae of the history; for example, the remarks on page 906 cast some interesting light on the way in which Russian and US algebraic topologists gradually became aware of the importance of the theory of formal group laws in complex cobordism, leading to Quillen’s seminal paper. Oddly, no mention is made of the further development of this based on observations of Morava which eventually led to the now ubiquitous chromatic viewpoint in Stable Homotopy Theory; indeed little mention is made of applications of complex cobordism and formal group theory in Stable Homotopy Theory beyond the early work of Novikov, except for the very contemporary results of Hill, Hopkins and Ravenel on the Kervaire invariant. Another notable omission is mention of Adams’ Chicago Notes which were very important in making the work of Novikov and Quillen accessible to topologists outwith the Soviet Union. It is probably fair to say that the inclusion and exclusion of topics reflects the author’s interests and personal involvement in the subject, but a more inclusive survey might be of value.
What is covered emphasises the geometric aspects including for example, multiplicative genera and group actions.
The extent of the mathematical ground covered is impressive, but it is probably not easy to learn details from this paper. A major difficulty is with the references which seem to be in a very unobvious order, making it hard to follow up sources.
There appear to be some unfortunate errors in the text, and the reviewer cautions the reader to check more detailed sources: for example the discussion of the James Splitting on page 914 seems to be missing some suspensions.

MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R77 Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism)
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