An introduction to torsion subcomplex reduction

TL;DR

This survey introduces Torsion Subcomplex Reduction (TSR), a technique that simplifies the computation of torsion in group cohomology, leading to new formulas and progress in various areas of geometric and algebraic topology.

Contribution

The paper presents TSR as a novel method for directly accessing reduced torsion subcomplexes, enabling explicit formulas and advancing understanding of cohomology and K-theory of specific groups.

Findings

Derived formulas for tetrahedral Coxeter groups' cohomology

Refined the Quillen conjecture using new torsion formulas

Proved Ruan's crepant resolution conjecture for Bianchi orbifolds

Abstract

This survey paper introduces to a technique called Torsion Subcomplex Reduction (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cell complexes. TSR enables one to skip machine computations on cell complexes, and to access directly the reduced torsion subcomplexes, which yields results on the cohomology of matrix groups in terms of formulas. TSR has already yielded general formulas for the cohomology of the tetrahedral Coxeter groups as well as, at odd torsion, of SL2 groups over arbitrary number rings. The latter formulas have allowed to refine the Quillen conjecture. Furthermore, progress has been made to adapt TSR to Bredon homology computations. In particular for the Bianchi groups, yielding their equivariant K-homology, and, by the Baum-Connes assembly map, the K-theory of their reduced C *-algebras. As a side application, TSR has allowed to provide dimension formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds, and to prove Ruan's crepant resolution conjecture for all complexified Bianchi orbifolds.