An index relation for the quilted Atiyah-Floer conjecture
David L. Duncan

TL;DR
This paper proves that the chain level gradings of instanton Floer groups and quilted Lagrangian Floer groups coincide for certain 3-manifolds, advancing the understanding of the Atiyah-Floer conjecture.
Contribution
It establishes the equality of gradings between two Floer theories, providing a key step towards the quilted Atiyah-Floer conjecture.
Findings
Gradings of instanton and quilted Floer groups are shown to agree.
Supports the conjecture relating different Floer theories for 3-manifolds.
Enhances the mathematical framework connecting gauge theory and symplectic geometry.
Abstract
Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. Each of these is equipped with a chain level grading. We show that the gradings agree.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Geometric Analysis and Curvature Flows
