Progress report on the telescope conjecture

Abstract

The Telescope Conjecture (made public in a lecture at Northwestern University in 1977) says that the vn–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.