Semi-simplicial Weil algebras and characteristic classes

Abstract

The purpose of this paper is the construction and coho- mological study of semi-simplici al models for the Weil algebra of a Lie algebra. The geometric context where the authors introduced these alge- bras is the construction of generalized characteristic classes for foliated bundles. There are two main aspects to the results of this paper. The first is the homological equivalence of all semi-simplicial Weil algebras even after passing to basic elements with respect to a subalgebra and to quotients by certain characteristic filtration ideals. The geometric con- sequence is that the generalized characteristic homomorphisms defined on these various complexes all have the same domain of universal generalized characteristic invariants. The second aspect is a comparison map from the ordinary Weil algebra to the semi-simplicial Weil algebra realizing a homology isomorphism after passing to basic elements with respect to a subalgebra and to quotients by characteristi c filtration ideals. The geo- metric consequence is a comparison of the characteristic class constructions on the various complexes considered, which is also of significance for ex- plicit computations.