Uniqueness of Massey Products on the Stable Homotopy of Spheres

The product on the stable homotopy ring of spheres π*scan be defined by composing, smashing or joining maps. Each of these three points of view is used in Section 2 to define Massey products on π*s. In fact we define composition and smash Massey products (x1, … , xt)where X1, … ,xt-1∈ π*s, xt∈ π*(E) and E is a spectrum. In Theorem 3.2, we prove that these three types of Massey products are equal. Consequently, a theorem which is easy to prove for one of these Massey products is also valid for the other two. For example, [3, Theorem 8.1] which relates algebraic Massey products in the Adams spectral sequence to Massey smash products in π*s is now also valid for Massey composition products in π*s

Detection of a new nontrivial family in the stable homotopy of spheres $ \mbf{\pi_{\ast}S} $

To determine the stable homotopy groups of spheres is one of the central problems in homotopy theory. Let $ A $ be the mod $ p $ Steenrod algebra and $S$ the sphere spectrum localized at an odd prime $ p $. In this article, it is proved that for $ p\geqslant 7 $, $ n\geqslant 4 $ and $ 3\leqslant s $, $ b_0 h_1 h_n \tilde{\gamma}_{s} \in Ext_A^{s+4,\ast}(\mathbb{Z}_p,\mathbb{Z}_p) $ is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element of order $ p $ in the stable homotopy groups of spheres $ \pi_{p^nq+sp^{2}q+(s+1)pq+(s-2)q-7}S $, where $ q=2(p-1 ) $.