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

Dain Kim, Nicholas Wilkins

TL;DR

The paper investigates the $\mathbb{Z}/p$-equivariant cohomology of the genus-zero Deligne–Mumford space $\overline{\mathcal{M}}_{0,1+p}$ with $\mathbb{F}_p$-coefficients, proving that the Serre spectral sequence for 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 $E_2$. Using localization, the authors show that torsion elements in the equivariant cohomology correspond to non-equivariant classes, implying that the only nontrivial $\mathbb{Z}/p$-equivariant operations arise as quantum Steenrod power operations. The main technical contribution is a detailed analysis of the $E_2$-page, fixed-point contributions, and the two differential-vanishing arguments that force collapse. Consequently, the paper concludes that, apart from non-equivariant Gromov–Witten-type operations, there are no exotic $u$-torsion equivariant operations at this level, providing a sharp description of the $\mathbb{Z}/p$-equivariant quantum operations for genus-zero moduli spaces. This advances understanding of equivariant quantum invariants and clarifies the role of fixed-point data in governing equivariant cohomology 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.

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

TL;DR

The paper investigates the -equivariant cohomology of the genus-zero Deligne–Mumford space with -coefficients, proving that the Serre spectral sequence for the fibration collapses at . Using localization, the authors show that torsion elements in the equivariant cohomology correspond to non-equivariant classes, implying that the only nontrivial -equivariant operations arise as quantum Steenrod power operations. The main technical contribution is a detailed analysis of the -page, fixed-point contributions, and the two differential-vanishing arguments that force collapse. Consequently, the paper concludes that, apart from non-equivariant Gromov–Witten-type operations, there are no exotic -torsion equivariant operations at this level, providing a sharp description of the -equivariant quantum operations for genus-zero moduli spaces. This advances understanding of equivariant quantum invariants and clarifies the role of fixed-point data in governing equivariant cohomology operations.

Abstract

We prove that the Serre spectral sequence of the fibration collapses at the page. We use this to prove that: for any element of the -equivariant cohomology with -coefficients of genus zero Deligne-Mumford space with marked points, if this element is torsion then it is non-equivariant. This concludes that the only "interesting" -equivariant operations are quantum Steenrod power operations.
Paper Structure (16 sections, 12 theorems, 39 equations)

This paper contains 16 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of paper
  4. Preliminaries
  5. The Deligne-Mumford space of genus zero
  6. Cohomology with local systems
  7. $\mathbb{Z}/p$-equivariant cohomology
  8. Serre spectral sequence
  9. Localization
  10. The structure of $H^*(\overline{\mathcal{M}}_{0, 1+p};\mathbb{F}_p)$ and the Serre spectral sequence
  11. The cohomology of $\overline{\mathcal{M}}_{0, 1+p}$ and its fixed point under $\mathbb{Z}/p$ action
  12. The $E_2$ page of the Serre spectral sequence
  13. Main Result
  14. The spectral sequence collapses
  15. The proof of \ref{['thm:main']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $\overline{\mathcal{M}}_{0, 1+p}$ be the Deligne-Mumford space of genus zero with $1+p$ marked points. Then the following map is injective: where $\rho$ is induced by forgetting equivariant parameters and $i^\ast$ by the inclusion $i \colon \overline{\mathcal{M}}_{0, 1+p}^{\text{fix}} \hookrightarrow \overline{\mathcal{M}}_{0, 1+p}$.

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5: Serre
  • Theorem 2.6: Quillen
  • Theorem 3.1: Etingof
  • ...and 14 more