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Cohomology at Infinity and the Well-Tempered Complex

Dylan Galt, Mark McConnell

TL;DR

The paper develops a finite-term framework to compute Hecke actions on the cohomology of the Borel-Serre boundary for $SL_n$ by extending the well-tempered complex to the boundary. It constructs a one-parameter, boundary-compatible family of neighborhoods and central tiles, yielding a deformation retraction from $\overline{X}\times[1,\tau_0]$ onto the well-tempered retract $\tilde{W}^+$ and a pair of boundary spectral sequences that encode cohomology at infinity. The core result is a sequence of cubical commutative diagrams tying together spectral sequences from prior work with the well-tempered framework, allowing the Hecke operator $T_a$ to be computed in finite terms via slices at critical temperaments. The framework provides practical algorithms for obtaining the action of Hecke operators on $H^*_{\Gamma}(\overline{X})$, with implications for interior cohomology, ghost classes, and Langlands-type questions related to Eisenstein series and cuspidal automorphic forms, and includes concrete computational strategies implemented in Sage and Haskell.

Abstract

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a method for computing in finite terms the action of Hecke operators on the equivariant cohomology of an arithmetic subgroup $Γ$ of the special linear group $SL_n$.

Cohomology at Infinity and the Well-Tempered Complex

TL;DR

The paper develops a finite-term framework to compute Hecke actions on the cohomology of the Borel-Serre boundary for by extending the well-tempered complex to the boundary. It constructs a one-parameter, boundary-compatible family of neighborhoods and central tiles, yielding a deformation retraction from onto the well-tempered retract and a pair of boundary spectral sequences that encode cohomology at infinity. The core result is a sequence of cubical commutative diagrams tying together spectral sequences from prior work with the well-tempered framework, allowing the Hecke operator to be computed in finite terms via slices at critical temperaments. The framework provides practical algorithms for obtaining the action of Hecke operators on , with implications for interior cohomology, ghost classes, and Langlands-type questions related to Eisenstein series and cuspidal automorphic forms, and includes concrete computational strategies implemented in Sage and Haskell.

Abstract

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a method for computing in finite terms the action of Hecke operators on the equivariant cohomology of an arithmetic subgroup of the special linear group .
Paper Structure (10 sections, 13 theorems, 33 equations)

This paper contains 10 sections, 13 theorems, 33 equations.

Table of Contents

  1. The Well-Rounded Retract
  2. The Borel-Serre Compactification
  3. Boundary Neighborhoods for a Fixed Set of Weights
  4. The Well-Tempered Complex
  5. The Borel-Serre Boundary
  6. A Family of Central Tiles
  7. Extending to the Boundary
  8. Cohomology at Infinity
  9. Hecke Correspondences
  10. Comments on Implementation

Key Result

Lemma 3.1

There exists a $t=(t_1,\cdots, t_{k-1})\in \mathscr{U}_{\mathscr{F},f}$, depending on $f$, such that for any $\rho\in\mathscr{U}_{\mathscr{F},f}$ satisfying $\rho\leq t$, the flag $\mathscr{M}'$ of successive minima for $\rho\cdot f$ contains $\mathscr{F}$.

Theorems & Definitions (32)

  • Definition 1.1: avner_cohomology_1997, Def. 2.3
  • Definition 1.2: avner_cohomology_1997, Def. 2.5
  • Lemma 3.1: avner_cohomology_1997, Lemma 7.3
  • Proposition 1: avner_cohomology_1997, Prop. 7.4
  • Proposition 2: avner_cohomology_1997, Prop. 7.5
  • Definition 4.1: mcconnell_computing_2020, Def. 4
  • Remark 1
  • Definition 5.1
  • Lemma 5.1
  • proof
  • ...and 22 more