Cohomology of Topologised Monoids

TL;DR

This paper extends fundamental group cohomology results to topologised monoids and various cohomology theories, using explicit cochain calculations instead of homological algebra.

Contribution

It provides a direct, cochain-based proof of key cohomology theorems for topologised monoids and multiple cohomology theories, avoiding homological algebra methods.

Findings

Established long exact sequences for these cohomologies

Classified torsors via first cohomology groups

Proved spectral sequences and decomposition results

Abstract

We prove standard results of group cohomology -- namely, existence of a long exact sequence, classification of torsors via the first cohomology group, Shapiro's lemma, the Hochschild-Serre spectral sequence, a decomposition of the cochain complex in the direct product case, and Jannsen's result on the recovery problem -- for cohomology theories such as continuous, analytic, bounded, and pro-analytic cohomology. We also prove these results for certain monoids. The cohomology groups considered here all have very concrete interpretations by means of a cochain complex. Therefore, we do not use methods of homological algebra, but explicit calculations on the level of cochains, using techniques dating back to Hochschild and Serre.