  We calculate the exact parity odd part of the effective action
($\Gamma_{odd}^{2d+1}$) for massive Dirac fermions in 2d+1 dimensions at finite
temperature, for a certain class of gauge field configurations. We consider
first Abelian external gauge fields, and then we deal with the case of a
non-Abelian gauge group containing an Abelian U(1) subgroup. For both cases, it
is possible to show that the result depends on topological invariants of the
gauge field configurations, and that the gauge transformation properties of
$\Gamma_{odd}^{2d+1}$ depend only on those invariants and on the winding number
of the gauge transformation.
