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 via Devinatz, Hopkins, and Smith in 1985. over the last ten years a couple of major advances were made in homotopy conception, and this publication fills a true desire 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 tips that lead as much as the theorems and their functions to homotopy conception. The publication starts with a few straight forward recommendations of homotopy idea which are had to kingdom the matter. This comprises such notions as homotopy, homotopy equivalence, CW-complex, and suspension. subsequent the equipment of complicated cobordism, Morava K-theory, and formal team legislation in attribute *p* are brought. The latter part of the ebook offers experts with a coherent and rigorous account of the proofs. It contains hitherto unpublished fabric at the spoil product and chromatic convergence theorems and on modular representations of the symmetric group.

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**Sample text**

7) logF (x) = i≥0 xp . pi Fn is obtained by reducing F mod p and tensoring with Fpn . Now an automorphism e of Fn is a power series e(x) over Fpn satisfying e(Fn (x, y)) = Fn (e(x), e(y)). 3. 8) Fn e(x) = i ei xp . i≥0 More details can be found in [Rav86, Appendix 2]. 3. 4 that the cohomology of the group Sn figures prominently in the stable homotopy groups of finite complexes. For future reference we will record some facts about this cohomology here. Proofs and more precise statements can be found in [Rav86, Chapter 6].

For the sake of simplicity we will ignore suspensions in the rest of this discussion. The map from W (1) to Y will be denoted by g1 . The complex W (1) has type 1 and therefore a periodic selfmap f1 : Σd1 W (1) → W (1) which induces a K(1)∗ -equivalence. Now we can ask whether g1 becomes null homotopic when composed with some iterate of f1 or not. 1) f∗ f∗ f∗ 1 1 1 [W (1), Y ]S∗ −→ [Σd1 W (1), Y ]S∗ −→ [Σ2d1 W (1), Y ]S∗ −→ ··· , which we denote by v1−1 [W (1), Y ]S∗ . Note that the second part of the periodicity theorem implies that this limit is independent of the choice of f1 .

For X = S 0 , this isomorphism is the identity. (iii) DDX X and [X, Y ]∗ ∼ = [DY, DX]∗ . (iv) For a homology theory E∗ , there is a natural isomorphism between Ek (X) and E −k (DX). , for finite spectra X and Y , D(X ∧ Y ) = DX ∧ DY . (vi) The functor X → DX is contravariant. The Spanier-Whitehead dual DX of a finite complex X is analogous to the linear dual V ∗ = Hom(V, k) of a finite dimensional vector space V over a field k. 1(i) is analogous to the 2. SPANIER-WHITEHEAD DUALITY 53 isomorphism Hom(V, W ) ∼ =V∗⊗W for any vector space W .

### Nilpotence and periodicity in stable homotopy theory by Douglas C. Ravenel

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