  Complexes of discrete distributional differential forms are introduced into
finite element exterior calculus. Thus we generalize a notion of Braess and
Sch\"oberl, originally studied for a posteriori error estimation. We construct
isomorphisms between the simplicial homology groups of the triangulation, the
discrete harmonic forms of the finite element complex, and the harmonic forms
of the distributional finite element complexes. As an application, we prove
that the complexes of finite element exterior calculus have cohomology groups
isomorphic to the de Rham cohomology, including the case of partial boundary
conditions. Poincar\'e-Friedrichs-type inequalities will be studied in a
subsequent contribution.
