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The Lannes–Zarati homomorphism and decomposable elements

Ngô A Tuấn

Algebraic & Geometric Topology 19 (2019) 1525–1539
Abstract

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() vanishes on decomposable elements in positive stems.

Keywords
spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra
Mathematical Subject Classification 2010
Primary: 55P47, 55Q45, 55S10, 55T15
References
Publication
Received: 2 May 2018
Revised: 19 September 2018
Accepted: 15 October 2018
Published: 21 May 2019
Authors
Ngô A Tuấn
Department of Mathematics
Vietnam National University, Hanoi
Hanoi
Vietnam