Cocycles de groupe pour GL$_n$ et arrangements d'hyperplans

TL;DR

This paper presents a topological framework unifying various Eisenstein cocycles, linking topological classes to algebraic meromorphic forms on hyperplane complements, and constructing three types of Sczech cocycles.

Contribution

It introduces a topological construction that serves as a common origin for different Eisenstein cocycles, connecting topology with algebraic geometry.

Findings

Unified topological source for Eisenstein cocycles

Construction of three types of Sczech cocycles

Connection between topological classes and meromorphic forms

Abstract

Many authors have constructed different, but related, linear group cocycles that are usually referred to as ``Eisenstein cocycles.'' The main goal of this work is to describe a topological construction that is a common source for all these cocycles. One interesting feature of this construction is that, starting from a purely topological class, it leads to the algebraic world of meromorphic forms on hyperplane complements in $n$-fold products of either the (complex) additive group, the multiplicative group or a (family of) elliptic curve(s). This yields the construction of three types of ``Sczech cocycles.''