The motivic Mahowald invariant

Abstract

The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over $Spec(\mathbb{C})$. We compute a motivic version of the $C_2$-Tate construction for various motivic spectra, and show that this construction produces "blueshift" in these cases. We use these computations to show that the Mahowald invariant of ${\eta }^i$, $i\ge 1$, is the first element in Adams filtration $i$ of the $w_1$-periodic families constructed by Andrews ~\cite{And14}. This provides an exotic periodic analog of Mahowald and Ravenel's computation ~\cite{MR93} that the classical Mahowald invariant of $2^i$, $i\ge 1$, is the first element in Adams filtration $i$ of the $v_1$-periodic families constructed by Adams ~\cite{Ada66}.

arXiv comment40 pages

References in corpus (10)

• A comparison of motivic and classical homotopy theories
• Motivic Hopf elements and relations
• Classical and motivic Adams charts
• Some root invariants at the prime 2
• Exotic Motivic Periodicities
• Motivic connective K-theories and the cohomology of A(1)
• Motivic modular forms from equivariant stable homotopy theory
• The motivic Adams vanishing line of slope 1/2
• A class of $2$-local finite spectra which admit a $v_2^1$-self-map
• The Motivic Cofiber of $\tau$

Cited by in corpus (5)

• Real motivic and $C_2$-equivariant Mahowald invariants
• R-motivic stable stems
• $tmf$-based Mahowald invariants
• Tate blueshift and vanishing for Real oriented cohomology
• The $v_1$-Periodic Region of the Complex-Motivic Ext