THE COHOMOLOGY OF INVARIANT VARIATIONAL BICOMPLEXES

Let pi: E --> M be a fiber bundle and let Gamma be an infinitesimal Lie transformation group acting on E. We announce various new results concerning the cohomology of the Gamma invariant variational bicomplex (Omega(Gamma)*,*(J(infinity)(E)),d(H), d(V)) and the associated Gamma invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups.