The fibre of the degree 3 map, Anick spaces and the double suspension

Abstract

Let S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$ . We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Infinite families of higher torsion in the homotopy groups of Moore spaces

“…2 .2 r C1 / cannot exist unless r > 1 since D 2 . 2 P 2nC1 .2// is stably indecomposable by Theorem 1.1(c).…”

Higher chromatic Thom spectra via unstable homotopy theory

“…Under the J -homomorphism BString ! BGL 1 .S/, this generator maps to 2 8 BGL 1 .S/ Š 7 S, so 2 mod decomposables at p D 2, and y 6 maps to t3 1 mod decomposables at p D 3.One corollary (using Remark 3.2.1) is the following: Corollary 3.2.22 As A -comodules, we have H .BI F p / Š Example 3.2.23 For simplicity, let us work at p D 2.…”

“…The first part of this conjecture would follow from Proposition 4.1.6 if the Cohen-Moore-Neisendorfer map were a Gray map. In[3], it is shown that the existence of p-primary elements of Kervaire invariant one would imply equivalences of the form BW p n 1 ' T 2p n C1 .p/.…”