# Squaring operations in the $RO(C_2)$-graded and real motivic Adams spectral sequences

## TL;DR

This paper develops a formula for differentials involving squaring operations in the $C_2$-equivariant and motivic Adams spectral sequences, enabling more precise computations in equivariant and motivic stable homotopy theory.

## Contribution

It establishes an $H_{ ext{infty}}$-structure on Adams towers and determines attaching maps for $C_2$-equivariant projective spaces, leading to new differential formulas.

## Key findings

- Derived a formula for $d_2(sq^i(x))$ differentials.
- Determined attaching maps for $C_2$-equivariant projective spaces.
- Provided sample computations illustrating the formulas.

## Abstract

In this paper we establish a formula for computing $d_2(sq^i(x))$ where $x$ is a permanent cycle in the $C_2$-equivariant Adams spectral sequence or the motivic Adams spectral sequence over $Spec(\mathbb{R})$. This requires establishing that the Adams towers have an $H_{\infty}$-structure as well as determining the attaching maps for $C_2$-equivariant projective spaces. The attaching maps of $C_2$-equivariant projective spaces can then be used to determine the coefficients of differentials in both the equivariant and motivic case. At the end some sample computations are given.