$tmf$-based Mahowald invariants

Abstract

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.

v2: 50 pages, 42 figures. Charts updated for readability. Some other minor corrections. To appear in Algebraic & Geometric Topology

References in corpus (2)

• The motivic Mahowald invariant
• Real motivic and $C_2$-equivariant Mahowald invariants

Cited by in corpus (1)

• The Real-oriented cohomology of infinite stunted projective spaces