The $\mathbb Z/p$-equivariant cohomology of genus zero Deligne-Mumford space with $1+p$ marked points

TL;DR

This paper proves the collapse of a spectral sequence for a specific equivariant cohomology of genus zero Deligne-Mumford space, showing that torsion elements are non-equivariant, and identifies quantum Steenrod power operations as the main equivariant operations.

Contribution

It establishes the collapse of the Serre spectral sequence for the $b Z/p$-equivariant cohomology of genus zero Deligne-Mumford space and characterizes torsion elements as non-equivariant, highlighting quantum Steenrod power operations.

Findings

Spectral sequence collapses at $E_2$ page.

Torsion elements in equivariant cohomology are non-equivariant.

Quantum Steenrod power operations are the only interesting equivariant operations.

Abstract

We prove that the Serre spectral sequence of the fibration $\overline{\mathcal M}_{0, 1+p} \to E\mathbb Z/p \times_{\mathbb Z/p} \overline{\mathcal M}_{0, 1+p} \to B \mathbb Z/p$ collapses at the $E_2$ page. We use this to prove that: for any element of the $\mathbb Z/p$-equivariant cohomology with $\mathbb F_p$-coefficients of genus zero Deligne-Mumford space with $1+p$ marked points, if this element is torsion then it is non-equivariant. This concludes that the only "interesting" $\mathbb Z/p$-equivariant operations are quantum Steenrod power operations.