AN ELEMENTARY AND DIRECT COMPUTATION OF COHOMOLOGY WITH AND WITHOUT A GROUP ACTION (Probability Symposium)

Abstract

Recently, we introduced a configuration space with interaction structure and a uniform local cohomology on it with co-authors in [1]. 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 [l]. 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.