$\mathit{tmf}$-based Mahowald invariants

TL;DR

This paper computes $ ext{tmf}$-based approximations to the $2$-primary homotopy $eta$-family, an infinite collection of periodic elements in stable homotopy groups, using spectral sequences and Mahowald invariant techniques.

Contribution

It introduces a novel approach combining spectral sequence analysis and Mahowald invariants to approximate the $eta$-family in stable homotopy groups.

Findings

Computed $ ext{tmf}$-based approximations to the $eta$-family.

Analyzed the Atiyah-Hirzebruch spectral sequence for Tate construction of $ ext{tmf}$.

Applied Behrens' filtered Mahowald invariant machinery.

Abstract

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate $\mathit{tmf}$-based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of $\mathit{tmf}$ with trivial $C_2$-action and Behrens' filtered Mahowald invariant machinery.