On the Bott periodicity, $J$-homomorphisms, transfer maps and $H_*Q_0S^{-n}$

TL;DR

This paper explores the structure of spherical classes in the homology of infinite loop spaces related to the sphere spectrum, using Bott periodicity, $J$-homomorphisms, and spectral sequences to advance understanding of the Curtis conjecture.

Contribution

It introduces new generators in homology via Bott periodicity and $J$-homomorphisms, and analyzes subalgebra structures and spherical classes in these contexts.

Findings

Curtis conjecture holds when restricted to the $J$ component.

Identified generators in $H_*(Q_0S^0; ext{Z}/p)$ using Bott periodicity.

Determined spherical classes in $H_*( ext{loop spaces } ext{J}; ext{Z}/2)$.

Abstract

The Curtis conjecture predicts that the only spherical classes in $H_*(Q_0S^0;\Z/2)$ are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan's decomposition $$Q_0S^0=J\times\cok J$$ where $J$ is the fibre of $\psi^q-1$ ($q=3$ at the prime $2$) and observe that the Curtis conjecture holds when we restrict to $J$. We then use the Bott periodicity and the $J$-homomorphism $SO\to Q_0S^0$ to define some generators in $H_*(Q_0S^0;\Z/p)$, when $p$ is any prime, and determine the type of subalgebras that they generate. For $p=2$ we determine spherical classes in $H_*(\Omega^k_0J;\Z/2)$. We determine truncated subalgebras inside $H_*(Q_0S^{-k};\Z/2)$. Applying the machinery of the Eilenberg-Moore spectral sequence we define classes that are not in the image of by the $J$-homomorphism. We shall make some partial observations on the algebraic structure of $H_*(\Omega^k_0\cok J;\Z/2)$. Finally, we shall make some comments on the problem in the case equivariant $J$-homomorphisms.