An elementary and direct computation of cohomology with and without a group action

Abstract

Recently, we introduced a configuration space with interaction structure and a uniform local cohomology on it with co-authors in arXiv:2009.04699. The notion is used to understand a common structure of infinite product spaces appeared in the proof of Varadhan's non-gradient method. For this, the cohomology of the configuration space with a group action is the main target to study, but the cohomology is easily obtained from that of the configuration space without a group action by applying a well-known property on the group cohomology. In fact, the analysis of the cohomology of the configuration space without a group action is the essential part of arXiv:2009.04699. In this article, we give an elementary and direct proof to obtain the cohomology of a space with a group action from that without a group action under a certain condition including the setting of the configuration space with interaction structure. In particular, no knowledge of group cohomology is required.

The manuscript was already published (without peer review) in RIMS Kokyuroku No.2177 in 2021