The Hurewicz image of the $\eta_i$ family, a polynomial subalgebra of $H_*\Omega_0^{2^{i+1}-8+k}S^{2^i-2}$

TL;DR

This paper calculates the Hurewicz image of Mahowald's family in stable homotopy groups, identifying spherical classes and their algebraic structures in certain loop space homologies, and discusses implications for the Curtis conjecture.

Contribution

It provides explicit calculations of spherical classes in loop space homologies and analyzes their algebraic structures, advancing understanding of the Curtis conjecture.

Findings

Identification of specific spherical classes in homology

Determination of algebraic structures of subalgebras

Insights into the Curtis conjecture and spherical classes in loop spaces

Abstract

We consider the problem of calculating the Hurewicz image of Mahowald's family $\eta_i\in{_2\pi_{2^i}^S}$. This allows us to identify specific spherical classes in $H_*\Omega_0^{2^{i+1}-8+k}S^{2^i-2}$ for $0\leqslant k\leqslant 6$. We then identify the type of the subalgebras that these classes give rise to, and calculate the $A$-module and $R$-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in $H_*Q_0S^0$, and relations with spherical classes in $H_*Q_0S^{-n}$.