The A-decomposability of the Singer construction

TL;DR

This paper proves a key conjecture in algebraic topology, showing that a specific algebraic transformation related to the Singer construction is trivial in positive degrees for certain parameters, advancing understanding of spherical classes.

Contribution

It proves the weak generalized algebraic spherical class conjecture, establishing the triviality of a natural transformation for s>2, which was previously unconfirmed.

Findings

The morphism is trivial on positive degree elements for s>2.

The condition s>2 is necessary, as shown by known spherical classes.

Advances understanding of the algebraic structure of spherical classes.

Abstract

Let $R_s M$ denote the Singer construction on an unstable module $M$ over the Steenrod algebra $A$ at the prime two; $R_s M$ is canonically a subobject of $P_s\otimes M$, where $P_s$ is the polynomial algebra on s generators of degree one. Passage to $A$-indecomposables gives the natural transformation $R_s M \rightarrow F \otimes_A (P_s \otimes M)$, which identifies with the dual of the composition of the Singer transfer and the Lannes-Zarati homomorphism. The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first named author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.