Katsumi Shimomura

The homotopy groups π∗(L2S0) at the prime 3

Abstract The homotopy groups π ∗ (L 2 S 0 ) of the L2-localized sphere are determined by studying the Bockstein spectral sequence. The results also indicate the homotopy groups π ∗ (L K(2) S 0 ) and we observe that the fiber of the localization map L2S30→LK(2)S0 is homotopic to Σ−2L1S30. Here S30 denotes the 3-completed sphere.

Note on beta elements in homotopy, and an application to the prime three case

Let S 0 (p) denote the sphere spectrum localized at an odd prime p. Then we have the first beta element β 1 ∈ π 2p 2 -2p-2 (S 0 (p) ); whose cofiber we denote by W. We also consider a generalized Smith-Toda spectrum V r characterized by BP * (V r ) = BP * /(p,v r l ). In this note, we show that an element of π * (V r Λ W) gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of β 9t+3 at the prime three to complete a conjecture of Ravenels: β s ∈ π 16s―6 (S 0 (3) ) exists if and only if s is not congruent to 4, 7 or 8 mod 9.

On the homotopy groups of E.n/-local spectra with unusual invariant ideals

Let E.n/ and T.m/ for nonnegative integers n and m denote the Johnson‐Wilson and the Ravenel spectra, respectively. Given a spectrum whose E.n/ ‐homology is E.n/ .T.m//=.v1;:::;vn 1/, then each homotopy group of it estimates the order of each homotopy group of LnT.m/. We here study the E.n/‐based Adams E2 ‐term of it and present that the determination of the E2 ‐term is unexpectedly complex for odd prime case. At the prime two, we determine the E1 ‐term for .L2T.1/=.v1//, whose computation is easier than that of .L2T.1// as we expect. 55Q99

E(2)-invertible spectra smashing with the Smith-Toda spectrum V(1) at the prime 3

Let L 2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E(2). A spectrum L 2 X is called invertible if there is a spectrum Y such that L 2 X A Y = L 2 S 0 . Hovey and Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L 2 S 0 at a prime p > 3. At the prime 3, it is shown, A relation between the Picard group of the E(n)-local homotopy category and E(n)-based Adams spectral sequence, that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L 2 S 0 . In this paper, we show the homotopy equivalence v 3 2 : Σ 48 L 2 V(1) ≃ V(1) A X for the Smith-Toda spectrum V(1). In the same manner as this, we also show the existence of the self-map β: Σ 144 L 2 V(1) → L 2 V(1) that induces v 9 2 on the E(2)*-homology.

Generalized Bousfield Lattices and a Generalized Retract Conjecture

In [1], Bousfield studied a lattice (Bousfield lattice) on the stable homotopy category of spectra, and in [5], Hovey and Palmieri made the retract conjecture on the lattice. In this paper we generalize the Bousfield lattice and the retract conjecture to the ones on a monoid. We also determine the structure of typical examples of them, which satisfy the generalized retract conjecture. In particular we give the structure of the Bousfield lattice of the stable homotopy category of harmonic spectra explicitly.

A beta family in the homotopy of spheres

Let p be a prime number greater than three. In the p-component of stable homotopy groups of spheres, Oka constructed a beta family from a v2-periodic map on a four cell complex. In this paper, we construct another beta family in the groups at a prime p greater than five from a v2-periodic map on an eight cell complex.

The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

The chromatic spectral sequence is introduced in (8) to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term E s,t (k) of the spectral sequence is an Ext group of BP∗BP-comodules. There are a sequence of Ext groups E s,t (n s) for non-negative integers n with E s,t 1 (0) = E

THE HOMOTOPY GROUPS OF THE L2-LOCALIZATION OF THE MODULO p MOORE SPECTRUM SMASHING WITH THE FIRST RAVENEL SPECTRUM

Let BP be the Brown-Peterson spectrum at an odd prime p, and L2 denote the Bousfield localization functor with respect to . The Ravenel spectrum T(1) is characterized by BP*(T(1)) = BP*[t1] on the primitive generator t1. In this paper, we determine the homotopy groups π*(L2M ∧ T(1)) for the mod p Moore spectrum M.