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

TL;DR

This paper introduces spectral invariants in equivariant monopole Floer homology for rational homology three-spheres, linking these invariants to geometric properties like scalar curvature obstructions.

Contribution

It develops an $oldsymbol{R}$-filtration and spectral invariant $ ho$ within equivariant monopole Floer homology, providing new tools to study geometric obstructions.

Findings

Spectral invariant $ ho$ obstructs positive scalar curvature metrics.

$oldsymbol{R}$-filtration integrates monopole Floer homology into persistent homology.

$ ho$ is related to the geometry of the underlying 3-manifold.

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.