Let be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism vanishes on classes of of Adams filtration greater than . Let denote the Lannes–Zarati homomorphism for the unstable –module . When , this homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the Lannes–Zarati homomorphism, , vanishes in any positive stem for and for any unstable –module .
We prove that, for an unstable –module of finite type, the Lannes–Zarati homomorphism, , vanishes on decomposable elements of the form in positive stems, where and with either , and , or , and . Consequently, we obtain a theorem proved by Hung and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for vanishes on decomposable elements in positive stems.
"The Lannes–Zarati homomorphism and decomposable elements." Algebr. Geom. Topol. 19 (3) 1525 - 1539, 2019. https://doi.org/10.2140/agt.2019.19.1525