Index of transverse Dirac operator and cohomotopy Seiberg-Witten invariant for codimension $4$ Riemannian foliation

Abstract

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 $H_b^1(M)\cap H^1(M,\mathbb{Z})$ is a lattice of $H_b^1(M)$, 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 $H_b^{\pm }(M)=1$, 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.