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Global methods in homotopy theory. (English) Zbl 0657.55008
Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987).
[For the entire collection see Zbl 0628.00011.]
This enjoyable paper gives a survey of the ideas which lie behind the recent solution of the nilpotence conjecture in stable homotopy theory. In section 1 the proof of the nilpotence theorem [E. S. Devinatz, M. J. Hopkins and J. H. Smith: Nilpotence and stable homotopy theory I, Ann. Math. II. Ser. 128, 207-241 (1988)] is explained, whereas section 2, 3 sketch refinements (complex cobordism replaced by Morava K- theories) and applications (to \(v_ n\)-self-maps) to be detailed by M. J. Hopkins and J. H. Smith in “Nilpotence and stable homotopy theory II” (to appear). Section 4 illustrates the underlying philosopy with an algebraic example of chain complexes over some ring. We cite from the last paragraph: “... We have now reached a vista where the algebra and the geometry are really the same thing: the study of periodicity in stable homotopy is the same as the study of the prime ideals of the sphere spectrum...”
Reviewer: E.Ossa

55P42 Stable homotopy theory, spectra
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55Q45 Stable homotopy of spheres
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology