$tmf$-based Mahowald invariants

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