tmf–based Mahowald invariants

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