The nilpotence and periodicity theorems in stable homotopy theory. (English) Zbl 0728.55003

Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189-190, Exp. No. 728, 399-428 (1990).
[For the entire collection see Zbl 0722.00001.]
This very readable article sketches the proof of the Nilpotence Theorem, proved by E. S. Devinatz, M. J. Hopkins and J. H. Smith [Ann. Math., II. Ser. 128, No.2, 207-241 (1988; Zbl 0673.55008)], and the Periodicity Theorem, proved by Hopkins and Smith in 1985, but still not published. This paper does an admirable job in the daunting task of explaining the statement and proofs of these results to a reader who has no specialized knowledge. It is highly recommended for graduate students who would like to learn about what are probably the most celebrated results in homotopy theory of the period 1985-1991. The first result says that in the stable homotopy category a self map of a finite complex is nilpotent if and only if it is nilpotent when viewed through the eyes of the complex bordism functor \(MU_*(-).\) The second result says that every noncontractible finite complex has a non-nilpotent self map, and this is asymptotically unique.


55P42 Stable homotopy theory, spectra
55Q10 Stable homotopy groups
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