Cohomology of the fundamental groups of toroidal groups

Abstract

We construct an isomorphism between the $\overline{\partial}$-cohomology and the cohomology of the fundamental groups of toroidal groups, and get a standard form of $p$-cocycles, which was given by Vogt [6] in case of $1$-cocycles. Using differential forms via the above isomorphism enables us to obtain new results in higher dimensional cases. An explicit isomorphism between the Čech cohomology and the cohomology of the fundamental groups of complex tori is given in [5] (p.14). Our results give a generalization of this isomorphism to toroidal groups.