On behavior of the Lannes-Zarati homomorphism

TL;DR

This paper constructs a chain-level representation of the Lannes-Zarati homomorphism for specific cohomology theories, demonstrating its triviality in certain degrees and analyzing its behavior in low homological degrees.

Contribution

It provides a novel chain-level construction of the Lannes-Zarati homomorphism and investigates its properties at specific degrees, extending understanding of its behavior.

Findings

The sixth Lannes-Zarati homomorphism for F_2 is trivial for t ≤ 114.

The method reveals the behavior of the homomorphism for *B at s  4.

New chain-level representation of the homomorphism is constructed.

Abstract

In this paper, we construct the chain level representation in the lambda algebra of the Lannes-Zarati homomorphism for $\mathbb{F}_2$ as well as for $\tilde{H}^*B\mathbb{Z}/2$. Using this construction, we show that the sixth Lannes-Zarati homomorphism for $\mathbb{F}_2$ is trivial at all stems $t$ for $t\leq 114$. Furthermore, using this method, we also investigate the behavior of the Lannes-Zarati homomorphis for $\tilde{H}^*B\mathbb{Z}/2$ at the homological $s\leq 4$.