The generalized algebraic conjecture on spherical classes

摘要

Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism H : p*( Q0X). H*(Q0X) vanishes on classes of p*(Q0X) of Adams filtration greater than 2. Let.s : Exts A( H * ( X), F2). (F2. A Rs H* ( X))* denote the sth Lannes-Zarati homomorphism for the unstable Amodule 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 RP8 vanishes in positive stems, while the first Lannes-Zatati homomorphism for any space is basically non-zero. We prove the algebraic conjecture for RP8 and RPn with s = 3, 4. We discuss the relation between the Lannes-Zarati homomorphisms for RP8 and S0. Consequently, the algebraic conjecture for X = S0 is re-proved with s = 3, 4, 5.