Nilpotence and periodicity in stable homotopy theory.

*(English)*Zbl 0774.55001
Annals of Mathematics Studies. 128. Princeton, NJ: Princeton University Press. xiv, 209 p. (1992).

The aim of this book is to present the results of the fascinating new developments in stable homotopy theory known as the chromatic theory. This theory came into life roughly 15 years ago. An essential point was the discovery of the chromatic spectral sequence [H. R. Miller, the author and W. S. Wilson, Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)], but the theory obtained its conceptual appearance and importance essentially through the author’s conjectures, published in the author, [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003)].

To formulate these conjectures, it will be convenient to work in the homotopy category \({\mathcal S}\) of \(p\)-local spectra at a fixed prime \(p\). The most important generalized cohomology theory on this category is given by the Brown-Peterson spectrum \(BP\) whose coefficients form a polynomial ring \(BP_ *=\mathbb{Z}_{(p)}[v_ 1,\dots,v_ n,\dots]\), where the generator \(v_ n\) has degree \(| v_ n|=2(p^ n-1)\). From it one derives the \(BP\)-module spectra \(E(n)\) and \(K(n)\) with coefficients \(E(n)_ *=\mathbb{Z}_{(p)}[v_ 1,\dots,v_ n,v_ n^{-1}]\) and \(K(n)_ *=\mathbb{F}_ p[v_ n,v_ n^{-1}]\); the latter is called the \(n\)-th Morava \(K\)-theory. Bousfield has shown [A. K. Bousfield, Topology 18, 257-281 (1979; Zbl 0417.55007)] that corresponding to any homology theory \(E\) on \({\mathcal S}\) there is a localization functor \(L_ E:{\mathcal S}\to{\mathcal S}\) turning \(E\)-homology equivalences into isomorphisms, together with a natural transformation \(l_{\mathcal S}\to L_ E\) preserving \(E\)-homology. Two spectra \(E\) and \(E'\) have the same Bousfield class if their localization functors are equivalent. It is customary to use the abbreviation \(L_ n=L_{E(n)}\) for localization with respect to \(E(n)\).

The intention of the chromatic theory is to understand the category \({\mathcal F}\subset{\mathcal S}\) of \(p\)-local finite complexes through a filtration related to the functors \(L_ n\). A finite \(p\)-local complex \(X\) is said to be of type \(n\) if \(K(i)_ *X=0\) for \(i<n\) and \(K(i)_ *X\neq 0\) for \(i\geq n\). This is equivalent to \(L_{n-1}X\simeq *\) and \(L_ nX\not\simeq *\). A selfmap of \(X\) is a map \(f:\Sigma^ qX\to X\) (where \(\Sigma^ q\) is \(q\)-fold suspension); \(f\) is called a \(v_ n\)- selfmap if \(K(i)_ *f=0\) for \(i\neq n\) whereas \(K(n)_ *f\) is a (nontrivial) isomorphism.

We shall now describe the main conjectures referred to above. All of these have been clarified by now, so we shall not refer to them as conjectures any more. The most basic one is the nilpotence theorem which states that a selfmap \(f:\Sigma^ qX\to X\) with \(X\in{\mathcal F}\) and \(BP_ *(f)=0\) must be nilpotent (in the sense that \(f\circ\cdots\circ f\) is homotopic to zero). This was proved in [E. S. Devinatz, M. J. Hopkins and J. H. Smith, Ann. Math., II. Ser., 128, 207-241 (1988; Zbl 0673.55008)]. The most important consequence is the periodicity theorem (proved in [M. J. Hopkins, Global methods in homotopy theory, Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)]. It states that a type \(n\) complex \(X\) has an asymptotically unique \(v_ n\)- selfmap; here “asymptotically” refers to taking iterated compositions of selfmaps. A further consequence is the thick category theorem which classifies the categories \({\mathcal C}\) of finite \(p\)-local spectra closed under cofibrations and retracts: apart from the trivial exceptions such a thick category \({\mathcal C}\) consists of all complexes of type \(n\) for some \(n=n({\mathcal C})\).

As a corollary, the Bousfield classes of finite \(p\)-local complexes are determined completely by their types; this is the class invariance theorem. Moreover, there is the so-called Boolean algebra theorem describing completely the Boolean algebra generated by these Bousfield classes and their complements. Finally, we have the chromatic convergence theorem which asserts that for a finite \(p\)-local complex \(X\) the localizations \(L_ nX\) converge to the homotopy type of \(X\). Without restricting to finite complexes one has the following two important results on the localization functors \(L_ n\). The first is the localization theorem stating that \(BP_ *(L_ nX)=(L_ nBP)_ *X\); since \(L_ nBP\sim v_ n^{-1}BP\) this is a very usable description. Secondly, there is the smashing theorem, that \(L_ nX\simeq X\wedge L_ nS^ 0\).

As remarked above, all these results were conjectured in [the author, loc.cit.] and partly proved there in particular cases. He also formulated a telescope conjecture, now known to be false in general.

With these preparations, we can now describe the contents of the book. The first part addresses itself to quite a general audience without assuming specialized knowledge in stable homotopy theory. This should be useful both to mathematicians from other fields who would like to get an overview over the current developments in homotopy theory and to the graduate student just starting in this area. In particular for the latter type of reader appendices \(A\) and \(B\) should be very useful as they provide a quick introduction into some more detailed material which would be required for a more profound understanding.

Anyway, the first 5 chapters together with chapter 7 make for a very readable and informative overview over the theory. Chapter 1 introduces the basic definitions and just states as the main results the nilpotence and periodicity theorems. In chapter 2 the basic notions are expanded somewhat more and the chromatic filtration is described. Thus in these two chapters the general idea of the chromatic theory is sketched. Chapter 3 introduces formal group laws and the complex cobordism functor; this is here presented as a functor taking values in the category of \(\Gamma\)-modules, where \(\Gamma\) is the group under composition of power series \(x+\sum_{i\geq 1}b_ ix^{i+1}\) with integer coefficients. Landweber’s prime ideal theorem and filtration theorem are quoted and the type-\(n\)-notion is introduced. After this the thick category theorem and its algebraic version for \(p\)-local \(\Gamma\)-modules can be formulated. Chapter 4 is somewhat more technical; it contains the definition of the Morava stabilizer groups (as automorphism groups of certain formal groups over \(\mathbb{F}_ p)\) and some results on their cohomology, the proofs, however, are deferred to a later chapter. In chapter 5 we find a short discussion of smash products of spectra and Spanier-Whitehead duality. Then it is shown how the thick category theorem follows from the nilpotence theorem. The reader not interested in the technical details should now perhaps skip the next chapter and go to chapter 7 where he is told the basic facts about localization and Bousfield classes. Also the class invariance and Boolean algebra theorems are derived, and the telescope conjecture is discussed.

The rest of the book requires a more thorough understanding of stable homotopy theory. A reader not familiar with this field who still wants to go on should perhaps consult first appendices \(A\) and \(B\). Appendix \(A\) contains more on spectra and generalized homology theories as well as an introduction to the Adams spectral sequence. In \(B\) we find the necessary information on complex cobordism with its interplay to formal group theory, and on the machinery around the Brown-Peterson spectrum.

We now return to the description of the main text. To complete the picture outlined so far there remain still several major steps to be taken. Chapter 6 is devoted to the proof of existence for a \(v_ n\)- selfmap on some finite \(p\)-local complex. This proof follows the original method of J. H. Smith which has still not appeared in print (but is available in preprint form). The method requires a detailed study of the classical Adams spectral sequence. Basically one shows that the vanishing of sufficiently many Margolis homology groups of \(H^*(Y)\) guarantees a good vanishing line for the Adams spectral sequence of the function spectrum of selfmaps of \(Y\). Following Smith one then constructs a suitable \(Y\) with the help of the modular representation theory of the symmetric group; this is explained in more detail in appendix \(C\).

The first part of chapter 8 describes \(L_ nBP\) and the proof of the localization theorem; this is taken from [the author, Algebraic topology and algebraic \(K\)-theory, Proc. Conf., Princeton, NJ (USA), Ann. Math. Studies 113, 168-179 (1987; Zbl 0707.55004)]. The rest of the chapter is devoted to the proofs of the smash product theorem and the chromatic convergence theorem. These are due to the author and M. J. Hopkins, but have not appeared in print before. They require Bousfields theory on the convergence of the general Adams spectral sequence (see [A. K. Bousfield, loc. cit.]) and a careful study of the cohomology of the Morava stabilizer groups with coefficients in the Morava \(K\)-theory of complex projective spaces. The relevant cohomological properties of profinite groups are also explained in the text. Finally, chapter 9 outlines the proof of the nilpotence theorem, following basically the original paper [E. S. Devinatz, M. J. Hopkins, J. H. Smith, loc. cit.].

Especially the second part of the book, providing full proofs for the important new results mentioned above, makes it a very useful and highly welcome addition to the literature. For those looking for first information on the chromatic theory in stable homotopy, the first part can be recommended as a rather complete but still easy-going overview over the subject.

A list of errata is now available by e-mail (under drav @ harpo.math.rochester.edu).

To formulate these conjectures, it will be convenient to work in the homotopy category \({\mathcal S}\) of \(p\)-local spectra at a fixed prime \(p\). The most important generalized cohomology theory on this category is given by the Brown-Peterson spectrum \(BP\) whose coefficients form a polynomial ring \(BP_ *=\mathbb{Z}_{(p)}[v_ 1,\dots,v_ n,\dots]\), where the generator \(v_ n\) has degree \(| v_ n|=2(p^ n-1)\). From it one derives the \(BP\)-module spectra \(E(n)\) and \(K(n)\) with coefficients \(E(n)_ *=\mathbb{Z}_{(p)}[v_ 1,\dots,v_ n,v_ n^{-1}]\) and \(K(n)_ *=\mathbb{F}_ p[v_ n,v_ n^{-1}]\); the latter is called the \(n\)-th Morava \(K\)-theory. Bousfield has shown [A. K. Bousfield, Topology 18, 257-281 (1979; Zbl 0417.55007)] that corresponding to any homology theory \(E\) on \({\mathcal S}\) there is a localization functor \(L_ E:{\mathcal S}\to{\mathcal S}\) turning \(E\)-homology equivalences into isomorphisms, together with a natural transformation \(l_{\mathcal S}\to L_ E\) preserving \(E\)-homology. Two spectra \(E\) and \(E'\) have the same Bousfield class if their localization functors are equivalent. It is customary to use the abbreviation \(L_ n=L_{E(n)}\) for localization with respect to \(E(n)\).

The intention of the chromatic theory is to understand the category \({\mathcal F}\subset{\mathcal S}\) of \(p\)-local finite complexes through a filtration related to the functors \(L_ n\). A finite \(p\)-local complex \(X\) is said to be of type \(n\) if \(K(i)_ *X=0\) for \(i<n\) and \(K(i)_ *X\neq 0\) for \(i\geq n\). This is equivalent to \(L_{n-1}X\simeq *\) and \(L_ nX\not\simeq *\). A selfmap of \(X\) is a map \(f:\Sigma^ qX\to X\) (where \(\Sigma^ q\) is \(q\)-fold suspension); \(f\) is called a \(v_ n\)- selfmap if \(K(i)_ *f=0\) for \(i\neq n\) whereas \(K(n)_ *f\) is a (nontrivial) isomorphism.

We shall now describe the main conjectures referred to above. All of these have been clarified by now, so we shall not refer to them as conjectures any more. The most basic one is the nilpotence theorem which states that a selfmap \(f:\Sigma^ qX\to X\) with \(X\in{\mathcal F}\) and \(BP_ *(f)=0\) must be nilpotent (in the sense that \(f\circ\cdots\circ f\) is homotopic to zero). This was proved in [E. S. Devinatz, M. J. Hopkins and J. H. Smith, Ann. Math., II. Ser., 128, 207-241 (1988; Zbl 0673.55008)]. The most important consequence is the periodicity theorem (proved in [M. J. Hopkins, Global methods in homotopy theory, Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)]. It states that a type \(n\) complex \(X\) has an asymptotically unique \(v_ n\)- selfmap; here “asymptotically” refers to taking iterated compositions of selfmaps. A further consequence is the thick category theorem which classifies the categories \({\mathcal C}\) of finite \(p\)-local spectra closed under cofibrations and retracts: apart from the trivial exceptions such a thick category \({\mathcal C}\) consists of all complexes of type \(n\) for some \(n=n({\mathcal C})\).

As a corollary, the Bousfield classes of finite \(p\)-local complexes are determined completely by their types; this is the class invariance theorem. Moreover, there is the so-called Boolean algebra theorem describing completely the Boolean algebra generated by these Bousfield classes and their complements. Finally, we have the chromatic convergence theorem which asserts that for a finite \(p\)-local complex \(X\) the localizations \(L_ nX\) converge to the homotopy type of \(X\). Without restricting to finite complexes one has the following two important results on the localization functors \(L_ n\). The first is the localization theorem stating that \(BP_ *(L_ nX)=(L_ nBP)_ *X\); since \(L_ nBP\sim v_ n^{-1}BP\) this is a very usable description. Secondly, there is the smashing theorem, that \(L_ nX\simeq X\wedge L_ nS^ 0\).

As remarked above, all these results were conjectured in [the author, loc.cit.] and partly proved there in particular cases. He also formulated a telescope conjecture, now known to be false in general.

With these preparations, we can now describe the contents of the book. The first part addresses itself to quite a general audience without assuming specialized knowledge in stable homotopy theory. This should be useful both to mathematicians from other fields who would like to get an overview over the current developments in homotopy theory and to the graduate student just starting in this area. In particular for the latter type of reader appendices \(A\) and \(B\) should be very useful as they provide a quick introduction into some more detailed material which would be required for a more profound understanding.

Anyway, the first 5 chapters together with chapter 7 make for a very readable and informative overview over the theory. Chapter 1 introduces the basic definitions and just states as the main results the nilpotence and periodicity theorems. In chapter 2 the basic notions are expanded somewhat more and the chromatic filtration is described. Thus in these two chapters the general idea of the chromatic theory is sketched. Chapter 3 introduces formal group laws and the complex cobordism functor; this is here presented as a functor taking values in the category of \(\Gamma\)-modules, where \(\Gamma\) is the group under composition of power series \(x+\sum_{i\geq 1}b_ ix^{i+1}\) with integer coefficients. Landweber’s prime ideal theorem and filtration theorem are quoted and the type-\(n\)-notion is introduced. After this the thick category theorem and its algebraic version for \(p\)-local \(\Gamma\)-modules can be formulated. Chapter 4 is somewhat more technical; it contains the definition of the Morava stabilizer groups (as automorphism groups of certain formal groups over \(\mathbb{F}_ p)\) and some results on their cohomology, the proofs, however, are deferred to a later chapter. In chapter 5 we find a short discussion of smash products of spectra and Spanier-Whitehead duality. Then it is shown how the thick category theorem follows from the nilpotence theorem. The reader not interested in the technical details should now perhaps skip the next chapter and go to chapter 7 where he is told the basic facts about localization and Bousfield classes. Also the class invariance and Boolean algebra theorems are derived, and the telescope conjecture is discussed.

The rest of the book requires a more thorough understanding of stable homotopy theory. A reader not familiar with this field who still wants to go on should perhaps consult first appendices \(A\) and \(B\). Appendix \(A\) contains more on spectra and generalized homology theories as well as an introduction to the Adams spectral sequence. In \(B\) we find the necessary information on complex cobordism with its interplay to formal group theory, and on the machinery around the Brown-Peterson spectrum.

We now return to the description of the main text. To complete the picture outlined so far there remain still several major steps to be taken. Chapter 6 is devoted to the proof of existence for a \(v_ n\)- selfmap on some finite \(p\)-local complex. This proof follows the original method of J. H. Smith which has still not appeared in print (but is available in preprint form). The method requires a detailed study of the classical Adams spectral sequence. Basically one shows that the vanishing of sufficiently many Margolis homology groups of \(H^*(Y)\) guarantees a good vanishing line for the Adams spectral sequence of the function spectrum of selfmaps of \(Y\). Following Smith one then constructs a suitable \(Y\) with the help of the modular representation theory of the symmetric group; this is explained in more detail in appendix \(C\).

The first part of chapter 8 describes \(L_ nBP\) and the proof of the localization theorem; this is taken from [the author, Algebraic topology and algebraic \(K\)-theory, Proc. Conf., Princeton, NJ (USA), Ann. Math. Studies 113, 168-179 (1987; Zbl 0707.55004)]. The rest of the chapter is devoted to the proofs of the smash product theorem and the chromatic convergence theorem. These are due to the author and M. J. Hopkins, but have not appeared in print before. They require Bousfields theory on the convergence of the general Adams spectral sequence (see [A. K. Bousfield, loc. cit.]) and a careful study of the cohomology of the Morava stabilizer groups with coefficients in the Morava \(K\)-theory of complex projective spaces. The relevant cohomological properties of profinite groups are also explained in the text. Finally, chapter 9 outlines the proof of the nilpotence theorem, following basically the original paper [E. S. Devinatz, M. J. Hopkins, J. H. Smith, loc. cit.].

Especially the second part of the book, providing full proofs for the important new results mentioned above, makes it a very useful and highly welcome addition to the literature. For those looking for first information on the chromatic theory in stable homotopy, the first part can be recommended as a rather complete but still easy-going overview over the subject.

A list of errata is now available by e-mail (under drav @ harpo.math.rochester.edu).

Reviewer: E.Ossa (Wuppertal)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P42 | Stable homotopy theory, spectra |

55T25 | Generalized cohomology and spectral sequences in algebraic topology |