Generalized Mahowald Invariants

The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. In this thesis, we study the Mahowald invariant in the settings of motivic stable homotopy theory over Spec(C) and Spec(R), as well as C2-equivariant stable homotopy theory. In the C2-equivariant setting, we prove an analog of Lin’s Theorem by adapting the motivic Singer construction developed by Gregersen to the C2-equivariant setting. We compute a motivic version of the Tate construction for various motivic spectra, and show that this construction produces 'blueshift' in these cases. In the complex motivic case, we use these computations to show that the Mahowald invariant of η^i , i ≥ 1, is the first element in Adams filtration i of the w1-periodic families constructed by Andrews. This provides an exotic periodic analog of Mahowald and Ravenel’s computation that the classical Mahowald invariant of 2^i , i ≥ 1, is the first element in Adams filtration i of the v1-periodic families constructed by Adams. We obtain similar results in both the real motivic and C2-equivariant settings. Finally, we also study the behavior of the Mahowald invariant under various functors between the motivic, equivariant, and classical stable homotopy categories.