By Douglas C. Ravenel

*Nilpotence and Periodicity in reliable Homotopy Theory* describes a few significant advances made in algebraic topology in recent times, centering at the nilpotence and periodicity theorems, which have been conjectured via the writer in 1977 and proved by way of Devinatz, Hopkins, and Smith in 1985. over the last ten years a few major advances were made in homotopy conception, and this booklet fills a true want for an updated textual content on that topic.

Ravenel's first few chapters are written with a normal mathematical viewers in brain. They survey either the guidelines that lead as much as the theorems and their purposes to homotopy conception. The publication starts off with a few effortless innovations of homotopy conception which are had to country the matter. This contains such notions as homotopy, homotopy equivalence, CW-complex, and suspension. subsequent the equipment of complicated cobordism, Morava K-theory, and formal staff legislation in attribute *p* are brought. The latter component to the ebook offers experts with a coherent and rigorous account of the proofs. It comprises hitherto unpublished fabric at the break product and chromatic convergence theorems and on modular representations of the symmetric group.

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A full subcategory C of CΓ is thick if it satisfies the following axiom: If 0 −→ M −→ M −→ M −→ 0 is a short exact sequence in CΓ, then M is in C if and only if M and M are. ) A full subcategory F of FH is thick if it satisfies the following two axioms: (i) If f X −→ Y −→ Cf is a cofibre sequence in which two of the three spaces are in F, then so is the third. (ii) If X ∨ Y is in F then so are X and Y . Thick subcategories were called generic subcategories by Hopkins in [Hop87]. 4. THICK SUBCATEGORIES 37 Using the Landweber filtration theorem, one can classify the thick subcategories of CΓ(p) .

Now suppose C ⊂ CΓ(p) is thick. If C = {0}, choose the largest n so that Cp,n ⊃ C. Then C ⊂ Cp,n+1 , and we want to show that C = Cp,n , so we need to verify that C ⊃ Cp,n . −1 M = 0 Let M be a comodule in C but not in Cp,n+1 . Thus vm −1 for m < n but vn M = 0. Choosing a Landweber filtration of M in CΓ, 0 = F0 M ⊂ F1 M ⊂ · · · ⊂ Fk M = M, all Fs M are in C, hence so are all the subquotients Fs M/Fs−1 M = Σds M U∗ /Ip,ms . Since vn−1 M = 0, we must have vn−1 (M U∗ /Ip,ms ) = 0 for some s, so some ms is no more than n.

1(i) is analogous to the 2. SPANIER-WHITEHEAD DUALITY 53 isomorphism Hom(V, W ) ∼ =V∗⊗W for any vector space W . 1(v) is analogous to the isomorphism (V ⊗ W )∗ ∼ = V ∗ ⊗ W ∗. The geometric idea behind Spanier-Whitehead duality is as follows. A finite spectrum X is the suspension spectrum of a finite CW-complex, which we also denote by X. The latter can always be embedded in some Euclidean space RN and hence in S N . Then DX is a suitable suspension of the suspension spectrum of the complement S N − X.