Mark V. Losik
The Cohomology of the Complex of G-Invariant Forms on G-Manifolds
The paper is published: Ann. Global Anal. Geom. 13 (1995) 323-338

MSC:

57R91 Equivariant algebraic topology of manifolds

Abstract: The cohomology of the complex of $G$-invariant forms for
arbitrary $G$-manifolds and, especially for a certain class of $G$-manifolds,
which are locally trivial fiber bundles over the orbit space, is considered.
The transgression in the differential graded algebra of basic elements for
tensor product of two identical Weil algebras of a group Lie $G$ with a
reductive Lie algebra is calculated and this is applied
to the calculation of the transgression of the cross product of principal
$G$-bundles over $G$. This allows to construct a convenient $DG$-algebra
whose cohomology coincides with the cohomology of the complex of $G$-invariant
forms on a $G$-manifold of the above class. In particular, for compact $G$
the generalization of the Cartan theorem on the cohomology of homogeneous
spaces is proved.

Keywords: G-manifold, fiber bundle, DG-algebra, cohomology, spectral sequence, transgression