On the form of potential spherical classes in $H_*Q_0S^0$

TL;DR

This paper investigates the structure of potential spherical classes in the homology of the infinite loop space of spheres, relating to a conjecture that only certain known elements produce spherical classes, and clarifies what alternative classes would look like if the conjecture fails.

Contribution

It characterizes the form of potential counterexamples to Curtis's conjecture, providing a target for future proofs or disproofs of the conjecture.

Findings

Determines the form of hypothetical spherical classes if Curtis's conjecture fails.

Sets conditions that any non-Hopf or non-Kervaire spherical class must satisfy.

Provides a framework for proving or disproving the conjecture based on these classes.

Abstract

This note is about spherical classes in $H_*Q_0S^0$. A conjecture, due to Ed. Curtis, predicts that only Hopf invariant one and Kervaire invariant one elements will give rise to spherical classes in $H_*Q_0S^0$. Yet, there has been no proof of this conjecture around. Assuming that this conjecture fails, there must exist some other spherical classes in $H_*Q_0S^0$. This note determines the form of these potential spherical classes, and sets the target for someone who wishes to prove the conjecture, in the sense that correctness of the Curtis conjecture will be the same as failure of any classes predicted in this paper being spherical.