Spherical classes detected by the algebraic transfer (Q1365443)

This article explains how the author's theorems play roles on the conjecture about spherical classes (CSC): There are no spherical classes in \(Q_0 S^0\) other than the Hopf invariant one and the Kervaire invariant one elements. The article consists of four theorems and seven conjectures including the above one. Five of the conjectures are weaker ones. These conjectures are stated by the language of Dickson algebras, the Lannes-Zarati homomorphism and Singer's lambda algebra. A theorem and two weaker conjectures imply the CSC. One of the other theorems is used to show that two of the conjectures are equivalent, another states a weaker version of a weaker conjecture, and the other says one of the conjectures holds for a special case. No proof is given in this note.