The 2-torsion in the Farrell--Tate cohomology of PSL(4,Z), and torsion subcomplex reduction via discrete Morse theory

TL;DR

This paper introduces a new, more efficient algorithm using discrete Morse theory to compute torsion subcomplexes, demonstrated by calculating the mod 2 Farrell-Tate cohomology of PSL(4,Z).

Contribution

It presents a novel implementation of torsion subcomplex reduction that improves simplicity and runtime for arithmetic group cohomology computations.

Findings

Algorithm achieves faster computation of torsion subcomplexes.

Successfully computes the mod 2 Farrell-Tate cohomology of PSL(4,Z).

Demonstrates effectiveness of discrete Morse theory in this context.

Abstract

In the present paper, we use discrete Morse theory to provide a new implementation of torsion subcomplex reduction for arithmetic groups. This leads both to a simpler algorithm as well as runtime improvements. To demonstrate the technique, we compute the mod 2 Farrell-Tate cohomology of PSL(4,Z).