# Real motivic and $C_2$-equivariant Mahowald invariants

## TL;DR

This paper extends the Mahowald invariant to $R$-motivic and $C_2$-equivariant contexts, revealing new periodic families and connecting classical, motivic, and equivariant homotopy theories.

## Contribution

It introduces the first periodic families in $R$-motivic and $C_2$-equivariant stable homotopy, linking these to classical $v_1$ and $w_1$-periodic families.

## Key findings

- Identifies $R$-motivic Mahowald invariants containing lifts of classical periodic elements.
- Establishes $C_2$-equivariant Mahowald invariants with similar periodic structures.
- Connects motivic and equivariant invariants to classical stable homotopy theory.

## Abstract

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