Liftings of forms to Weil bundles and the exterior derivative

Jacek D/ebecki¹

¹ Instytut Matematyki Uniwersytet Jagielloński Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 95 (2009) no. 3, pp. 289-300

Résumé

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.

Détail

DOI : 10.4064/ap95-3-7

Keywords: previous paper have given complete description linear liftings p forms n dimensional manifolds q forms where weil functor non negative integers except establish formulas connecting liftings exterior derivative forms these formulas contain boundary operator which enables define homology weil algebra study under condition acyclic finally compute kernels images boundary operators weil algebras mathbb these algebras acyclic