The exterior algebra for Wiemann manifolds

Abstract

We discuss the theory of infinite-dimensional manifolds from the point of view of establishing a widely applicable framework for generalization of the finite-dimensional Hodge theory. The principal result is the development of an exterior algebra based on a weakened definition of differentiation, so that “ C ∞” partitions of unity are available for paracompact manifolds modelled on arbitrary real separable Banach spaces. We prove a Poincaré lemma for our new notion of exterior differentiation, and go on to discuss the relationship of the exterior derivative with current research efforts toward the definition of an infinite-dimensional Laplace-Beltrami operator.