By Nigel Ray, Grant Walker
J. Frank Adams had a profound impact on algebraic topology, and his paintings maintains to form its improvement. The foreign Symposium on Algebraic Topology held in Manchester in the course of July 1990 was once devoted to his reminiscence, and almost the entire world's best specialists took half. This quantity paintings constitutes the court cases of the symposium; the articles contained the following variety from overviews to stories of labor nonetheless in growth, in addition to a survey and entire bibliography of Adam's personal paintings. those court cases shape an immense compendium of present examine in algebraic topology, and person who demonstrates the intensity of Adams' many contributions to the topic. This moment quantity is orientated in the direction of homotopy conception, the Steenrod algebra and the Adams spectral series. within the first quantity the topic is principally risky homotopy conception, homological and express.
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Extra resources for Adams Memorial Symposium on Algebraic Topology: Volume 2
Likewise, 7r* in cohomology shows it in fact to be bipolynomial. For r odd, the composite shows that i* is a monomorphism into an exterior algebra which also induces a monomorphism on indecomposables. Hence H*(BP(n)') is itself exterior. The K(n) cohomology result now follows from a simple application of the Atiyah-Hirzebruch spectral sequence. The sequence collapses as it is all in even dimensions and there are no multiplicative extension problems. The element vn E 7r2(pn_1)(BP(n)) gives rise to maps BP(n)r - BP(n)r_2(pn-1) (8) which in homology represent o-multiplying by [vn].
As in , or , this leads to a universal coefficient theorem. 7 Theorem. 1 ExtC7ZT(KCRTX' 7rCRTG) >-a [ X G] -HomcTT(K; RTX, 7rCRTG) 1 This sequence need not be splittable, since it gives a nontrivial exten- sion of Z/2 by Z/2 when X = FEMZ/2 and G = KO. 8 Corollary. Two spectra X and Y are quasi KO-equivalent if and only if K* RTX = K; RTY in CRT. 9 Theorem. For each Anderson exact module Al E CRT, there exists a spectrum X with Ii; RTX - M in CRT. Since 7r; RT(KO A X) = K; RTX , the above results combine to give the promised classification of KO-module spectra.
1)r+2(pi-1) j>n We deduce: Corollary 7. For r sufficiently small the Hopf algebra H*(BP(n)') is bipolynomial if r is even and exterior if r is odd. In particular, for r both even and small, K(n)*(BP(n)r) is a formal power series algebra. Hunton : Detruncating Morava K-theory 39 Proof. Theorem 6 gives a factorisation of the identity map BP(n)', BP;. --* BP(n);. for r sufficiently small. ). Likewise, 7r* in cohomology shows it in fact to be bipolynomial. For r odd, the composite shows that i* is a monomorphism into an exterior algebra which also induces a monomorphism on indecomposables.
Adams Memorial Symposium on Algebraic Topology: Volume 2 by Nigel Ray, Grant Walker