Towards a nonabelian cohomology of forms

Abstract

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotopy invariants--for manifolds. The corresponding Hodge theory yields finiteness of nonabelian Betti numbers. A genralized Poincar\'{e} lemma, along with a Poincar\'{e} duality, a Mayer-Vietoris, and a particularly empowered Bockstein makes our cohomology computable. Bockstein also allows us to relate (nonabelian) diffeomorphism invariants to (abelian) homotopy invariants.