The generalized algebraic conjecture on spherical classes

Abstract

Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism$H: \pi _{*}(Q_0 X) \rightarrow H_{*}(Q_0 X)$vanishes on classes of$\pi _* (Q_0 X)$of Adams filtration greater than 2. Let $\varphi _s: {\rm Ext} _{\mathcal {A}}^{s}(\widetilde{H}^*(X), \mathbb {F}_2) \rightarrow {(\mathbb {F}_2 \otimes _{{\mathcal {A}}}R_s\widetilde{H}^*(X))}^*$ denote the sth Lannes–Zarati homomorphism for the unstable ${\mathcal {A}}$-module $\widetilde{H}^*(X)$. This homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the sth Lannes–Zarati homomorphism vanishes in any positive stem for $s>2$ and any CW-complex X. We construct a chain level representation for the Lannes–Zarati homomorphism by means of modular invariant theory. We show the commutativity of the Lannes–Zarati homomorphism and the squaring operation. The second Lannes–Zarati homomorphism for $\mathbb {R}\mathbb {P}^{\infty }$ vanishes in positive stems, while the first Lannes-Zatati homomorphism for any space is basically non-zero. We prove the algebraic conjecture for $\mathbb {R}\mathbb {P}^{\infty }$ and $\mathbb {R}\mathbb {P}^{n}$ with $s=3$, 4. We discuss the relation between the Lannes–Zarati homomorphisms for $\mathbb {R}\mathbb {P}^{\infty }$ and $S^0$. Consequently, the algebraic conjecture for $X=S^0$ is re-proved with $s=3$, 4, 5.