The Lannes–Zarati homomorphism and decomposable elements

Ngô, Tuấn¹

¹ Department of Mathematics, Vietnam National University, Hanoi, Hanoi, Vietnam

Algebraic and Geometric Topology, Tome 19 (2019) no. 3, pp. 1525-1539

Résumé

Let X be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism H : π∗(Q0X) → H∗(Q0X) vanishes on classes of π∗(Q0X) of Adams filtration greater than 2. Let φsM: ExtAs(M, F2) → (F2 ⊗ARsM)∗ denote the sth Lannes–Zarati homomorphism for the unstable A–module M. When M = H̃∗(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, φsM, vanishes in any positive stem for s > 2 and for any unstable A–module M.

We prove that, for M an unstable A–module of finite type, the sth Lannes–Zarati homomorphism, φsM, vanishes on decomposable elements of the form αβ in positive stems, where α ∈ ExtAp(F2, F2) and β ∈ ExtAq(M, F2) with either p ≥ 2, q > 0 and p + q = s, or p = s ≥ 2, q = 0 and stem(β) > s − 2. Consequently, we obtain a theorem proved by Hưng and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for H̃∗(ℝℙ∞) vanishes on decomposable elements in positive stems.

Détail

DOI : 10.2140/agt.2019.19.1525

Classification : 55P47, 55Q45, 55S10, 55T15
Keywords: spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra