The Farrell--Tate and Bredon homology for PSL\_4(Z) via cell subdivisions

TL;DR

This paper introduces algorithms for subdividing cell complexes to compute Farrell--Tate and Bredon (co)homology of PSL_4(Z), enabling new calculations for small primes and classifying spaces.

Contribution

It presents novel algorithms for cell subdivision ensuring cell stabilizers fix cells pointwise, facilitating homology computations for arithmetic groups.

Findings

Computed Farrell--Tate cohomology for small primes

Calculated Bredon homology for classifying spaces

Provided efficient subdivision algorithms for cell complexes

Abstract

We provide some new computations of Farrell--Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell--Tate or Bredon homology, one needs cell complexes wherecell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL4(Z) provides computations of Farrell--Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.