Nilpotence and stable homotopy theory. I.

*(English)*Zbl 0673.55008During the (roughly) last twenty years there has been a considerable development in the theoretical understanding of stable homotopy theory. Two main reasons for this can be singled out. On the algebraic and more computational side we find the discovery of periodic phenomena, based on the connections to formal group theory through complex cobordism theory MU and the Adams-Novokiv spectral sequence. On the more structural side the localization techniques of Bousfield have provided a means to classify generalized cohomology theories according to their relative strength. D. C. Ravenel [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003)] combined these two strands of ideas into a global vision of stable homotopy theory as stratified by different levels of periodicity. The open questions about this picture were formulated as a collection of 8 conjectures which have shaped and stimulated the subject.

The present paper (together with its companion [M. Hopkins, Proc. Symp. Durham 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)] which stresses the main ideas and theoretical consequences) certainly is one of the highlights in this development. It establishes and improves the most central of Ravenel’s conjectures which accordingly must now be named the nilpotence theorem. It comes in three versions:

Theorem 1: (i) Let R be a ring spectrum. The kernel of the MU-Hurewicz homomorphism \(MU_*: \pi_*R\to MU_*R\) consists of nilpotent elements. (ii) Let f: \(F\to X\) be a map from a finite spectrum to an arbitrary spectrum. If \(1_{MU}\wedge f\) is null homotopic, the n-fold smash product \(f\wedge...\wedge f\) is null homotopic for some n. (iii) Let \(...\to X_ n\to^{f_ n}X_{n+1}\to..\). be a sequence of spectra, where \(X_ n\) is \(c_ n\)-connected. If \(MU_*f_ n=0\) and if \(c_ n\geq mn+b\) for some m and b, then the telescope \(_ n\to X_ n\) is contractible.

Part (i) of the theorem is an easy consequence of part (ii), obtained by taking \(F=S^ 0\), \(X=R\). Also part (iii) is derived (in section 4 of the paper) from (ii) and the existence of some torsion-free (p-local) finite complexes with good Adams spectral sequence properties; these are constructed in (still unpublished) work by the third author.

In section 1 it is then shown that part (ii) of the theorem is implied by theorem 2, the latter being actually nothing else then part (i) of the nilpotence theorem for the case of a connective associative ring spectrum R of finite type. The remainder of section 1 contains a global overview over the proof of theorem 2.

The proof itself is obtained by filtering the MU-spectrum through Thom- spectra X(n) corresponding to the canonical vector bundles over SU(n). From here on one works locally at a fixed prime. Using the James (p- local) filtration of \(\Omega S^{2n-1}\), the spectra X(n) are again filtered by Spectra \(G_ k(n)\) with \(G_ 0(n)=X(n-1)\). Section 2 of the paper furnishes the induction step from X(n) to some \(G_ k(n)\) by a vanishing line argument in the X(n)-based Adams spectral sequence. Section 3 finally provides the reduction from \(G_ k(n)\) to \(G_{k- 1}(n)\). This is the most difficult part. The bulk of section 3 goes into the construction of p-local Wang-sequence-type cofibration maps for fibrations over \(J_{p-1}(S^{2np^{k-1}})\). Here (Proposition 3.33) \(G_ k(n)\) is related to the total space and \(G_{k-1}(n)\) to the fibre of such a fibration. An obstruction to the desired reduction remains only in the mapping telescope of the Wang-sequence-maps. But from the homological structure of \(\Omega^ 2S^{2m+1}\) it can finally be shown (Proposition 3.27) that this obstruction is controlled by mod-p-homology.

The present paper (together with its companion [M. Hopkins, Proc. Symp. Durham 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)] which stresses the main ideas and theoretical consequences) certainly is one of the highlights in this development. It establishes and improves the most central of Ravenel’s conjectures which accordingly must now be named the nilpotence theorem. It comes in three versions:

Theorem 1: (i) Let R be a ring spectrum. The kernel of the MU-Hurewicz homomorphism \(MU_*: \pi_*R\to MU_*R\) consists of nilpotent elements. (ii) Let f: \(F\to X\) be a map from a finite spectrum to an arbitrary spectrum. If \(1_{MU}\wedge f\) is null homotopic, the n-fold smash product \(f\wedge...\wedge f\) is null homotopic for some n. (iii) Let \(...\to X_ n\to^{f_ n}X_{n+1}\to..\). be a sequence of spectra, where \(X_ n\) is \(c_ n\)-connected. If \(MU_*f_ n=0\) and if \(c_ n\geq mn+b\) for some m and b, then the telescope \(_ n\to X_ n\) is contractible.

Part (i) of the theorem is an easy consequence of part (ii), obtained by taking \(F=S^ 0\), \(X=R\). Also part (iii) is derived (in section 4 of the paper) from (ii) and the existence of some torsion-free (p-local) finite complexes with good Adams spectral sequence properties; these are constructed in (still unpublished) work by the third author.

In section 1 it is then shown that part (ii) of the theorem is implied by theorem 2, the latter being actually nothing else then part (i) of the nilpotence theorem for the case of a connective associative ring spectrum R of finite type. The remainder of section 1 contains a global overview over the proof of theorem 2.

The proof itself is obtained by filtering the MU-spectrum through Thom- spectra X(n) corresponding to the canonical vector bundles over SU(n). From here on one works locally at a fixed prime. Using the James (p- local) filtration of \(\Omega S^{2n-1}\), the spectra X(n) are again filtered by Spectra \(G_ k(n)\) with \(G_ 0(n)=X(n-1)\). Section 2 of the paper furnishes the induction step from X(n) to some \(G_ k(n)\) by a vanishing line argument in the X(n)-based Adams spectral sequence. Section 3 finally provides the reduction from \(G_ k(n)\) to \(G_{k- 1}(n)\). This is the most difficult part. The bulk of section 3 goes into the construction of p-local Wang-sequence-type cofibration maps for fibrations over \(J_{p-1}(S^{2np^{k-1}})\). Here (Proposition 3.33) \(G_ k(n)\) is related to the total space and \(G_{k-1}(n)\) to the fibre of such a fibration. An obstruction to the desired reduction remains only in the mapping telescope of the Wang-sequence-maps. But from the homological structure of \(\Omega^ 2S^{2m+1}\) it can finally be shown (Proposition 3.27) that this obstruction is controlled by mod-p-homology.

Reviewer: E.Ossa

##### MSC:

55P42 | Stable homotopy theory, spectra |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

55Q10 | Stable homotopy groups |

55Q45 | Stable homotopy of spheres |