Edited by Nigel Ray
Edited by Grant Walker
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:January 2010
Online ISBN:9780511526312
Paperback ISBN:9780521421539
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511526312.007
Subjects: Geometry and Topology
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The Telescope Conjecture (made public in a lecture at Northwestern University in 1977) says that the υn–periodic homotopy of a finite complex of type n has a nice algebraic description. It also gives an explicit description of certain Bousfield localizations. In this paper we outline a proof that it is false for n = 2 and p ≥ 5. A more detailed account of this work will appear in [Rav]. In view of this result, there is no longer any reason to think it is true for larger values of n or smaller primes p.
In Section 1 we will give some background surrounding the conjecture. In Section 2 we outline Miller's proof of it for the case n = 1 and p > 2. This includes a discussion of the localized Adams spectral sequence. In Section 3 we describe the difficulties in generalizing Miller's proof to the case n = 2 We end that section by stating a theorem (3.5) about some differentials in the Adams spectral sequence, which we prove in Section 4. This material is new; I stated the theorem in my lecture at the conference, but said nothing about its proof. In Section 5 we construct the parametrized Adams spectral sequence, which gives us a way of interpolating between the Adams spectral sequence and the Adams–Novikov spectral sequence. We need this new machinery to use Theorem 3.5 to disprove the Telescope Conjecture. This argument is sketched in Section 6.
pp. i-iv
pp. v-vi
pp. vii-viii
Contents of Volume 1 : Read PDF
pp. ix-x
Programme of one-hour invited lectures : Read PDF
pp. xi-xi
Programme of contributed lectures : Read PDF
pp. xii-xii
Programme of Posters : Read PDF
pp. xiii-xiv
Participants in the Symposium : Read PDF
pp. xv-xvi
Addresses of Contributors : Read PDF
pp. xvii-xxiv
1 - Progress report on the telescope conjecture : Read PDF
pp. 1-22
2 - On K*-local stable homotopy theory : Read PDF
pp. 23-34
3 - Detruncating Morava K-theory : Read PDF
pp. 35-44
4 - On the p-adic interpolation of stable homotopy groups : Read PDF
pp. 45-54
5 - Some remarks on υ1 -periodic homotopy groups : Read PDF
pp. 55-72
6 - The unstable Novikov spectral sequence for Sp(n), and the power series sinh−1(x) : Read PDF
pp. 73-86
7 - Unstable Adams spectral sequence charts : Read PDF
pp. 87-118
8 - On a certain localization of the stable homotopy of the space XΓ : Read PDF
pp. 119-130
9 - Cooperations in elliptic homology : Read PDF
pp. 131-144
10 - Completions of G-spectra at ideals of the Burnside ring : Read PDF
pp. 145-178
11 - Theorems of Poisson, Euler and Bernouilli on the Adams spectral sequence : Read PDF
pp. 179-186
12 - Algebras over the Steenrod algebra and finite H-spaces : Read PDF
pp. 187-202
13 - The boundedness conjecture for the action of the Steenrod algebra on polynomials : Read PDF
pp. 203-216
14 - Representations of the homology of BV and the Steenrod algebra I : Read PDF
pp. 217-234
15 - Generic representation theory and Lannes' T-functor : Read PDF
pp. 235-262
16 - Some chromatic phenomena in the homotopy of MSp : Read PDF
pp. 263-280
17 - On a conjecture of Mahowald concerning bordism with singularities : Read PDF
pp. 281-290
18 - Topological gravity and algebraic topology : Read PDF
pp. 291-308