Real motivic and $C_2$-equivariant Mahowald invariants

Abstract

We generalize the Mahowald invariant to the $\mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i\equiv 2,3\mathrm{mod}4$, 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.

arXiv commentv2: 45 pages, substantially revised from v1. To appear in the Journal of Topology

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Cited by in corpus (1)

• $tmf$-based Mahowald invariants