Monopole Floer homology for codimension-3 Riemannian foliation

Abstract

In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\bar{HM}(M,F,\mfs;\Gamma),~\hat{HM}(M,F,\mfs;\Gamma), ~\widecheck{HM}(M,F,\mfs;\Gamma)$, for each transverse \spinc structure $\mfs$, where $\Gamma$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the Novikov ring on basic monopole Floer homologies.