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In this paper, we give a systematic study of Seiberg-Witten theory on closedoriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$.Under a certain topological condition, we construct the basic Seiberg-Witteninvariant 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 thesehomologies are independent of the bundle-like metric and generic perturbation.The major difference between the basic monopole Floer homologies and the oneson manifolds is the necessity to use the Novikov ring on basic monopole Floerhomologies.

Monopole Floer homology for codimension-3 Riemannian foliation