  Theorem. Let M be a compact, connected, oriented smooth Riemannian n-manifold
with non-empty boundary. Then the cohomology of the complex (Harm*(M),d) of
harmonic forms on M is given by the direct sum H^p(Harm*(M),d) = H^p(M;R) +
H^(p-1)(M;R) for p=0,1,...,n.
  When M is a closed manifold, a form is harmonic if and only if it is both
closed and co-closed. In this case, all the maps in the complex (Harm*(M),d)
are zero, and so H^p(Harm*(M),d) = Harm^p(M) = H^p(M;R) according to the
classical theorem of Hodge.
  By contrast, when M is connected and has non-empty boundary, it is possible
for a p-form to be harmonic without being both closed and co-closed. Some of
these, which are exact, although not exterior derivatives of harmonic
p-1-forms, represent the "echo" of the ordinary p-1-dimensional cohomology
within the p-dimensional harmonic cohomology that appears in the above theorem.
