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$\mathrm{tmf}$–based Mahowald invariants

### J D Quigley

Algebraic & Geometric Topology 22 (2022) 1789–1839
##### Abstract

The $2$–primary homotopy $\beta$–family, defined as the collection of Mahowald invariants of Mahowald invariants of ${2}^{i}$ for $i\ge 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. We calculate $\mathrm{tmf}$–based approximations to this family. Our calculations combine an analysis of the Atiyah–Hirzebruch spectral sequence for the Tate construction of $\mathrm{tmf}$ with trivial ${C}_{2}$–action and Behrens’ filtered Mahowald invariant machinery.

##### Keywords
tmf, Mahowald invariant, root invariant, Greek letter elements
##### Mathematical Subject Classification 2010
Primary: 55P42
Secondary: 55Q45, 55Q51