Spherical classes in some finite loop spaces of spheres

TL;DR

This paper verifies Eccles conjecture for certain finite loop spaces of spheres by classifying spherical classes in their homology, showing they only arise from Hopf invariant one elements, thus advancing understanding of spherical classes in loop spaces.

Contribution

It determines spherical classes in homology of specific finite and iterated loop spaces of spheres, confirming Eccles conjecture in these cases, and relates two conjectures in the field.

Findings

Spherical classes in $igcup_{d} ext{Homology}( ext{loop space of spheres})$ only from Hopf invariant one elements.

Complete classification of spherical classes in $ ext{Homology}( ext{single, double, triple loop spaces of spheres})$.

Verification of Eccles conjecture for these finite loop spaces.

Abstract

Working at the prime $2$, Curtis conjecture predicts that, in positive dimensions, spherical classes in $H_*QS^0$ only arise from Hopf invariant one and Kervaire invariant one elements. Eccles conjecture states that, in positive dimensional, for a path connected space $X$, a class in $H_nQX$ is spherical if its either stably spherical or it arises from a stable map $S^n\to X$ which is detected by a primary operation in its mapping cone. (i) We use Hopf invariant one result to verify Eccles conjecture on some finite loop spaces of spheres, namely, for $X=S^k$ , we completely determine spherical classes in $H_*(\Omega^dS^{k+d};\mathbb{Z}/2)$ for specific values of $d$, $k$ showing that a spherical classes in these cases only do arise from Hopf invariant one elements. (ii) We completely determine spherical classes in homology of single, double, and triple loop spaces of spheres, namely $\Omega S^{n+1}$, $\Omega^2 S^{n+2}$, and $\Omega^3S^{n+3}$ with $n>0$. These computations, verify Eccles conjecture on the finite loop spaces that we have considered. We also record some observations on the relation between two conjectures. The latter conjecture for $X=S^k$, with $k>0$, provides some evidence for the former to be true.