  It is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form
gauge theory provides an example of (i) a class of field theoretical models for
the Hodge theory, and (ii) a possible candidate for the quasi-topological field
theory (q-TFT). Despite many striking similarities with some of the key
topological features of the two $(1 + 1)$-dimensional (2D) free Abelian (and
self-interacting non-Abelian) gauge theories, it turns out that the 4D free
Abelian 2-form gauge theory is {\it not} an exact TFT. To corroborate this
conclusion, some of the key issues are discussed. In particular, it is shown
that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form
Abelian gauge theory obey the recursion relations that are reminiscent of the
exact TFTs but the Lagrangian density of this theory is not found to be able to
be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as
is the case with the {\it topological} 2D free Abelian (and self-interacting
non-Abelian) gauge theories.
