Monopole Floer homology for codimension-3 Riemannian foliation

TL;DR

This paper develops a new version of monopole Floer homology for 3-codimensional Riemannian foliations, extending Seiberg-Witten theory to foliated manifolds and introducing Novikov ring coefficients.

Contribution

It constructs basic Seiberg-Witten invariants and monopole Floer homologies for foliated manifolds, showing their independence from metrics and perturbations, and adapting the theory with Novikov ring coefficients.

Findings

Defined basic monopole Floer homologies for foliations

Proved invariance under metric and perturbation changes

Extended Floer homology to foliated manifolds using Novikov ring

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.