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Homology operations for gravity algebras

Tommaso Rossi

TL;DR

The paper computes the Σ_n-equivariant homology H_*^{Σ_n}(M_{0,n+1};F_p) and its sign-twisted variant, establishing these groups as generators of gravity-algebra homology operations and linking the homotopy quotient (M_{0,n+1})_{Σ_n} to B_n/Z(B_n). It develops two complementary strategies: a Serre spectral-sequence approach with BV-differentials for the trivial action, and a fibrewise/labelled configuration-space framework for the sign representation, reducing calculations to configuration spaces in the plane and its punctured variants. The main contributions include explicit descriptions and bases for H_*(B_n/Z(B_n);F_p) and H_*(B_n/Z(B_n);F_p(±1)), and a systematic construction of even- and odd-degree homology operations on gravity algebras from these equivariant classes. Together, these results yield concrete tools to understand gravity-algebra operations and clarify the connection between moduli-space homology, braid-group quotients, and the BV/Δ structure implicit in Dyer–Lashof-type operations.

Abstract

Let $\mathcal{M}_{0,n+1}$ be the moduli space of genus zero Riemann surfaces with $n+1$ marked points. In this paper we compute $H_*^{Σ_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p)$ and $H_*^{Σ_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p(\pm 1))$ for any $n\in\mathbb{N}$ and any prime $p$, where $\mathbb{F}_p(\pm 1)$ denotes the sign representation of the symmetric group $Σ_n$. The interest in these homology groups is twofold: on the one hand classes in these equivariant homology groups parametrize homology operations for gravity algebras. On the other hand the homotopy quotient $(\mathcal{M}_{0,n+1})_{Σ_n}$ is a model for the classifying space for $B_n/Z(B_n)$, the quotient of the braid group $B_n$ by its center.

Homology operations for gravity algebras

TL;DR

The paper computes the Σ_n-equivariant homology H_*^{Σ_n}(M_{0,n+1};F_p) and its sign-twisted variant, establishing these groups as generators of gravity-algebra homology operations and linking the homotopy quotient (M_{0,n+1})_{Σ_n} to B_n/Z(B_n). It develops two complementary strategies: a Serre spectral-sequence approach with BV-differentials for the trivial action, and a fibrewise/labelled configuration-space framework for the sign representation, reducing calculations to configuration spaces in the plane and its punctured variants. The main contributions include explicit descriptions and bases for H_*(B_n/Z(B_n);F_p) and H_*(B_n/Z(B_n);F_p(±1)), and a systematic construction of even- and odd-degree homology operations on gravity algebras from these equivariant classes. Together, these results yield concrete tools to understand gravity-algebra operations and clarify the connection between moduli-space homology, braid-group quotients, and the BV/Δ structure implicit in Dyer–Lashof-type operations.

Abstract

Let be the moduli space of genus zero Riemann surfaces with marked points. In this paper we compute and for any and any prime , where denotes the sign representation of the symmetric group . The interest in these homology groups is twofold: on the one hand classes in these equivariant homology groups parametrize homology operations for gravity algebras. On the other hand the homotopy quotient is a model for the classifying space for , the quotient of the braid group by its center.
Paper Structure (20 sections, 29 theorems, 68 equations, 1 figure)

This paper contains 20 sections, 29 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strategy of the proof (Theorem \ref{['thm: teorema 1 intro']}):
  3. Strategy of the proof (Theorem \ref{['thm: teorema 2 intro']}):
  4. Plan for the paper:
  5. Future directions:
  6. Acknowledgments:
  7. The Gravity operad
  8. Homology operations for gravity algebras
  9. Equivariant operations
  10. Homotopy models for $(\mathcal{M}_{0,n+1})_{\Sigma_n}$
  11. Operations for even degree classes
  12. Preliminares
  13. Equivariant cohomology
  14. Labelled configuration spaces
  15. Computation of $H_*^{\mathbb{Z}/p}(C_n(\mathbb{C});\mathbb{F}_p)$ when $n=0,1$ mod $p$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $n\in \mathbb{N}$ and $p$ a prime number. Then:

Figures (1)

  • Figure 1: This picture shows how the homeomorphism $f:C_{15}(\mathbb{C})^{\mathbb{Z}/5}\to C_3(\mathbb{C}^*)$ works.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Getzler, Getzler93
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Getzler Getzler94
  • Definition 2.5
  • Remark 3.1
  • ...and 58 more