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

2025/07/21 by Alexander D. Rahm, Bui, Anh Tuan, Anh Tuan Bui +3
Computer Science · Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2507.15368

openalex publication_date 2025/07/21 · openalex created_date 2026/04/07 · openalex updated_date 2026/07/28

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).

Citations

• Morse Theory for Complexes of Groups
• The Farrell--Tate and Bredon homology for PSL_4(Z) via cell subdivisions
• Accessing the cohomology of discrete groups above their virtual cohomological dimension
• The homological torsion of PSL₂ of the imaginary quadratic integers
• On the cohomology of congruence subgroups of SL₄ (\Z)
• Quelques calculs de la cohomologie de GL_N(Z) et de la K-theorie de Z

Related

• Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois\n representations
• The Farrell--Tate and Bredon homology for PSL_4(Z) via cell subdivisions
• On the Tate spectrum of tmf at the prime 2
• Torsion primes in loop space homology
• Torsion in cohomology and dimensional reduction