Integral cohomology of certain Picard modular surfaces

TL;DR

This paper presents a method to compute the integral cohomology of certain Picard modular surfaces associated with congruence subgroups, with implementations for specific imaginary quadratic fields.

Contribution

It introduces a novel computational approach for integral cohomology of Picard modular surfaces and applies it to explicit cases.

Findings

Computed integral cohomology for Gamma in Q(i) and Q(sqrt(-3))

Developed a practical method for cohomology calculations of locally symmetric spaces

Provided new data on the topology of Picard modular surfaces

Abstract

Let Gamma be a congruence subgroup of the Picard modular group of an imaginary number field k, and let D be the associated symmetric space. We describe a method to compute the integral cohomology of the locally symmetric space Gamma\D. The method is implemented for the case k=Q(i) and k=Q(sqrt(-3)), and the cohomology is computed for various Gamma.