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In the stable homotopy groups of spheres, we have Greek letter elements due to J. F. Adams [2], L. Smith [12] and H. Toda [13]. Here we study the nontriviality of certain products of the first alpha element, the first and the second beta elements and a gamma element in the homotopy groups.

In [1], Miller, Ravenel and Wilson defined generalized beta elements in the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [5], Oka showed that the beta elements of the form βtp2/r for positive integers t and r survives to the stable homotopy groups at a prime p > 3, when r ≤ 2p − 2 and r ≤ 2p if t > 1. In this paper, we expand the condition so that βtp2/r for r ≤ p2 − 3 survives to the stable homotopy groups.

Let V(0) denote the mod p Moore spectrum and L2 denote the Bousfield localization functor with respect to v2−1BP. The E2-term E2∗(L2V(0)) of the Adams–Novikov spectral sequence for computing homotopy groups π∗(L2V(0)) is E2∗(L2V(0))=ExtΓ∗(A,A) over the Hopf algebroid (A,Γ)=(E(2)∗,E(2)∗E(2)) for the Johnson–Wilson spectrum E(2), and obtained immediately from the chromatic E1-term H∗M11=ExtΓ∗(A,A/(v1∞)) for the generator v1 of A=Z/p[v1,v2±1]. In this paper, we determine the module structure of the E2-term E2∗(L2V(0)) at the prime 2 after determining H∗M11.

Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: $\al_1\ga_t$, $\be_2\ga_t$, $\lrk{\al_1,\al_1,\be_{p/p}^p}\ga_t\be_1$ and $\lrk{\be_1,p,\ga_t}$ for $t$ with $p\nmid t(t^2-1)$ for a prime number $p>5$., 11 pages

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.

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.