The ℤ∕p-equivariant cohomology of the genus-zero Deligne–Mumford space with 1 + p marked points

Kim, Dain; Wilkins, Nicholas

doi:10.2140/agt.2025.25.4341

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

Dain Kim and Nicholas Wilkins

Algebraic & Geometric Topology 25 (2025) 4341–4356

DOI: 10.2140/agt.2025.25.4341

Abstract

We prove that the Serre spectral sequence of the fibration ${\bar{ℳ}}_{0, 1 + p} \to E ℤ ∕ p \times_{ℤ ∕ p} {\bar{ℳ}}_{0, 1 + p} \to B ℤ ∕ p$ collapses at the $E_{2}$ -page. We use this to prove that a torsion element of the $ℤ ∕ p$ -equivariant cohomology with $𝔽_{p}$ -coefficients of genus-zero Deligne–Mumford space with $1 + p$ marked points is lifted from nonequivariant cohomology. We conclude that the only “interesting” $ℤ ∕ p$ -equivariant operations on quantum cohomology are quantum Steenrod power operations.

Keywords

Deligne–Mumford space, quantum Steenrod operations, equivariant cohomology

Mathematical Subject Classification

Primary: 53D45, 55N25, 55N91, 55T10

References

Publication

Received: 30 April 2024

Revised: 28 August 2024

Accepted: 16 September 2024

Published: 29 October 2025

Authors

Dain Kim
Department of Mathematics MIT Cambridge, MA United States

Nicholas Wilkins
Max Planck Institute for Mathematics Bonn Germany

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Volume 25, issue 7 (2025)