Unstable towers in the odd primary homotopy groups of spheres

Abstract

Abstract.The unstable elements in filtration 2 of the unstable Novikov spectral sequence are computed. These elements are shown to survive to elements in the homotopy groups of spheres which are related to Im J. The computation is applied to determine the Hopf invariants of compositions of Im / and the exponent of certain sphere bundles over spheres.

On nontriviality of certain homotopy groups of spheres

“…But this is not possible by (1). We conclude that τ * (α k−1 (2p − 1)) = 0 by the long exact sequence in (2), and so…”

“…The following statement seems to be known. For example, there is a discussion of this result at the end of page 535 in [2]. However, we were not able to find an explicit reference to this statement and give here a proof.…”

“…Here I : π 2p+2k(p−1)−1 (Q 3 2 ) → π 2p+2k(p−1)+2 (S 4p+1 ). Suppose that p * (γ ′ ) = 0, then γ ′ ∈ im{H (2) : π 2(p−1)(k+1)+4 (S 5 ) → π 2(p−1)(k+1)+1 (Q 3 2 ).} In this case, we get…”

“…Suppose that α k−1 (2p − 1) ∈ im{π * : π 2(p−1)k+2 (S 2p+1 ) → π 2(p−1)k (S 2p−1 )}. Then the element Σ 2 α k−1 (2p − 1) = α k−1 (2p + 1) is p-divisible by (2), hence its stable image is p-divisible. But this is not possible by (1).…”