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2019 The Lannes–Zarati homomorphism and decomposable elements
Ngô A Tuấn
Algebr. Geom. Topol. 19(3): 1525-1539 (2019). DOI: 10.2140/agt.2019.19.1525

Abstract

Let X be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism H : π ( Q 0 X ) H ( Q 0 X ) vanishes on classes of π ( Q 0 X ) of Adams filtration greater than 2 . Let φ s M : Ext A s ( M , F 2 ) ( F 2 A R s M ) denote the s th Lannes–Zarati homomorphism for the unstable A –module M . When M = H ̃ ( X ) , this homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the s th Lannes–Zarati homomorphism, φ s M , 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 s th Lannes–Zarati homomorphism, φ s M , vanishes on decomposable elements of the form α β in positive stems, where α Ext A p ( F 2 , F 2 ) and β Ext A q ( M , F 2 ) 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 Hung and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for H ̃ ( ) vanishes on decomposable elements in positive stems.

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Ngô A Tuấn. "The Lannes–Zarati homomorphism and decomposable elements." Algebr. Geom. Topol. 19 (3) 1525 - 1539, 2019. https://doi.org/10.2140/agt.2019.19.1525

Information

Received: 2 May 2018; Revised: 19 September 2018; Accepted: 15 October 2018; Published: 2019
First available in Project Euclid: 29 May 2019

zbMATH: 07078609
MathSciNet: MR3954290
Digital Object Identifier: 10.2140/agt.2019.19.1525

Subjects:
Primary: 55P47, 55Q45, 55S10, 55T15

Rights: Copyright © 2019 Mathematical Sciences Publishers

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