We discuss the BRST cohomology and exhibit a connection between the Hodge
decomposition theorem and the topological properties of a two dimensional free
non-Abelian gauge theory having no interaction with matter fields. The
topological nature of this theory is encoded in the vanishing of the Laplacian
operator when equations of motion are exploited. We obtain two sets of
topological invariants with respect to BRST and co-BRST charges on the two
dimensional manifold and show that the Lagrangian density of the theory can be
expressed as the sum of terms that are BRST- and co-BRST invariants.Comment: (1+11) pages, LaTeX, no figure