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...”

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

##### MSC:

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 |