Unstable Homotopy Theory Surrounding the Fibre of the pth Power Map on Loop Spaces of Spheres

In this thesis, we study the fibre of the $p^\text{th}$ power map on loop spaces of spheres with a view toward obtaining homotopy decompositions. For the prime $p=2$, we give an explicit decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the squaring map on the triple loop space of the $17$-sphere, building on work of Campbell, Cohen, Peterson and Selick, who showed that such decompositions are only possible for $S^5$, $S^9$ and $S^{17}$. This induces a splitting of the mod $2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2 \colon S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. For odd primes, we find that the decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong odd primary Kervaire invariant problem. Using this unstable homotopy theoretic interpretation of the Kervaire invariant problem together with what is presently known about the $3$-primary stable homotopy groups of spheres, we give a new decomposition of $\Omega S^{55}\{3\}$. We relate the $2$-primary decompositions above to various Whitehead products in the homotopy groups of mod $2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2,4,8$ or $16$. As an application of our $3$-primary decomposition, we prove two new cases of a longstanding conjecture which states that the fibre of the double suspension is a double loop space.Ph.D