Algebraic redshift in the $C_2$-equivariant Adams spectral sequence

TL;DR

This paper investigates $v_n$-periodic phenomena in $C_2$-equivariant stable homotopy using the Adams spectral sequence, constructing classes, defining torsion notions, and establishing algebraic Mahowald invariants with applications to motivic contexts.

Contribution

It introduces new methods to detect and analyze $v_n$-periodic classes in the $C_2$-equivariant Adams spectral sequence, extending classical splitting and invariant concepts.

Findings

Nonzero $v_n$ classes imply nonzero $v_{n-1}$ classes in related Ext groups.

Algebraic Mahowald invariants contain classes corresponding to higher $v_n$ powers.

Results extend to real motivic settings.

Abstract

We study $v_n$-periodic phenomena in $C_2$-equivariant stable homotopy through the lens of the $C_2$-equivariant Adams spectral sequence at the prime 2. In particular, we construct/detect certain classes related to powers of the $v_n$ generators of $\pi_*(BP)$ in the cohomology of certain finitely generated subalgebras $A^{C_2}(m)$ of the $C_2$-equivariant Steenrod algebra. We define the notion of classes in $\text{Ext}_{A^{C_2}}(\underline{H}^\star, \underline{H}^\star)$ being $v_n$-periodic or $v_n$-torsion and exhibit a chromatic filtration by showing that $v_n$-torsion classes are also $v_k$-torsion for $0\le k < n.$ We also promote the Lin-Davis-Mahowald-Adams splitting of Ext of the suitable version of ``$R P_{-\infty}^\infty$" to the $C_2$-equivariant setting and use this to define appropriate algebraic versions of Mahowald's root invariant. We establish that whenever a class corresponding to a power of $v_{n}$ is nonzero in $ \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star),$ then the same power of $v_{n-1}$ is also nonzero in $ \text{Ext}_{A^{C_2}(m-1)}(\underline{H}^\star, \underline{H}^\star),$ and its algebraic Mahowald invariant $M_m^{C_2-alg}(v_{n-1}^{2^f}) \subset \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star)$ contains class(es) corresponding to $v_n^{2^f}.$ Real motivic versions of these results hold as well.