The cohomology of the Steenrod algebra and the mod $p$ Lannes-Zarati homomorphism

TL;DR

This paper computes low-degree Ext groups over the Steenrod algebra and explores the properties of the Lannes-Zarati homomorphism for odd primes, advancing understanding of algebraic topology structures.

Contribution

It provides explicit calculations of Ext groups for the Steenrod algebra at low degrees and analyzes the behavior of the Lannes-Zarati homomorphism for odd primes.

Findings

Computed Ext groups for s ≤ 1 over the Steenrod algebra.

Analyzed the behavior of the Lannes-Zarati homomorphism at low degrees.

Enhanced understanding of algebraic structures in stable homotopy theory.

Abstract

In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $\varphi_3^{\mathbb{F}_p}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$.