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

## TL;DR

This paper explores the homotopy decomposition of certain fibre spaces related to degree maps on spheres, linking it to the Kervaire invariant problem and providing new decompositions and insights into longstanding conjectures.

## Contribution

It establishes the equivalence between the homotopy decomposition problem and the strong p-primary Kervaire invariant problem, and provides new decompositions for specific cases, including p=3.

## Key findings

- Proves the converse of the implication relating decomposition to Kervaire invariant elements.
- Shows the decomposition problem is equivalent to the strong p-primary Kervaire invariant problem.
- Provides new homotopy decompositions for -primary cases, including 55 for p=3.

## Abstract

Let $S^{2n+1}\{p\}$ denote the homotopy fibre of the degree $p$ self map of $S^{2n+1}$. For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$ admits a nontrivial loop space decomposition if and only if $n=1$ or $p$. Indecomposability in all but these dimensions was obtained by showing that a nontrivial decomposition of $\Omega S^{2n+1}\{p\}$ 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 $\Omega S^{2n+1}\{p\}$ 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 $\theta_3$ to give a new decomposition of $\Omega S^{55}\{3\}$ analogous to Selick's decomposition of $\Omega S^{2p+1}\{p\}$ 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.