  For closed manifolds endowed with a Riemannian foliation of codimension $4$,
one can define a transversal Seiberg-Witten map. We show that there is a finite
dimensional approximation for such a map. By such a method and under the
condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we
can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic
cohomological group is of zero dimension, we can give an estimate for the index
of transversal Dirac operator of a foliated spin structure. Furthermore, under
the condition that $H^\pm_b(M)=1$, we show the vanishing of the index of the
transverse Dirac operator. This gives a topological condition for the vanishing
of the index of the transverse Dirac operator.
