Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres

Abstract

In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $\mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an $\mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology $HM$. This $\mathbf{R}$-filtration fits $HM$ into a persistent homology theory, from which one can define a numerical quantity called the spectral invariant $\rho$. The spectral invariant $\rho$ is tied with the geometry of the underlying manifold. The main result of the papers shows that $\rho$ provides an obstruction to the existence of positive scalar curvature metric on a ribbon homology cobordism.