Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

In a previous paper we have given a complete description of linear liftings of $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on $T^AM$, where $T^A$ is a Weil functor, for all non-negative integers $n$, $p$ and $q$, except the case $p=n$ and $q=0$. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra~$A$. We next study the case $p=n$ and $q=0$ under the condition that $A$ is acyclic. Finally, we compute the kernels and the images of the boundary operators for the Weil algebras ${\mathbb D}^r_k$ and show that these algebras are acyclic.