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Register a new user$mathit{tmf}$-based Mahowald invariants
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Authors
J.D. Quigley
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The $2$-primary homotopy $beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate $mathit{tmf}$-based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of $mathit{tmf}$ with trivial $C_2$-action and Behrens filtered Mahowald invariant machinery.
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We generalize the Mahowald invariant to the $mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i equiv 2,3 mod 4$, we show that the $mathbb{R}$-motivic Mahowald invariant of $(2+rho eta)^i in pi_{0,0}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Adams classical $v_1$-periodic families, and for all $i > 0$, we show that the $mathbb{R}$-motivic Mahowald invariant of $eta^i in pi_{i,i}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Andrews $mathbb{C}$-motivic $w_1$-periodic families. We prove analogous results about the $C_2$-equivariant Mahowald invariants of $(2+rho eta)^i in pi_{0,0}^{C_2}(S^{0,0})$ and $eta^i in pi_{i,i}^{C_2}(S^{0,0})$ by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the $mathbb{R}$-motivic and $C_2$-equivariant settings.
The motivic Mahowald invariant was introduced in cite{Qui19a} and cite{Qui19b} to study periodicity in the $mathbb{C}$- and $mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of characteristic not two and use it to study periodicity in the $F$-motivic stable stems. In particular, we construct lifts of some of Adams classical $v_1$-periodic families cite{Ada66} and identify them as the motivic Mahowald invariants of powers of $2+rho eta$.
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