Families of metrics with positive scalar curvature on spectral sequence cobordisms

TL;DR

This paper constructs families of metrics with positive scalar curvature on cobordisms in monopole Floer spectral sequences, simplifying the analysis of differentials by eliminating irreducible solutions for certain link configurations.

Contribution

It introduces a method to produce positive scalar curvature metrics on specific cobordisms, reducing the complexity of counting solutions in monopole Floer spectral sequences.

Findings

Positive scalar curvature metrics eliminate irreducible solutions

Applicable to all configurations for T(2,n) torus knots

Includes configurations with exactly two 1-handle attachments

Abstract

We study families of metrics on the cobordisms that underlie the differential maps in Bloom's monopole Floer spectral sequence, a spectral sequence for links in $S^3$ whose $E^2$ is the Khovanov homology of the link, and which abuts to the monopole Floer homology of the double branched cover of the link. The higher differentials in the spectral sequence count parametrized moduli spaces of solutions to Seiberg-Witten equations, parametrized over a family of metrics with asymptotic behaviour corresponding to a configuration of unlinks with 1-handle attachments. For a class of configurations, we construct families of metrics with the prescribed behaviour, such that each metric therein has positive scalar curvature. The positive scalar curvature implies that there are no irreducible solutions to the Seiberg-Witten equations and thus, when the spectral sequences are computed with these families of metrics, only reducible solutions must be counted. The class of configurations for which we construct these families of metrics includes all configurations that go into the spectral sequence for $T(2,n)$ torus knots, and all configurations that involve exactly two 1-handle attachments.