$SO(3)$-Floer homology of 3-manifolds with boundary 1

TL;DR

This paper explores the relationship between Lagrangian Floer homology and Donaldson theory Floer homology for 3-manifolds with boundary, focusing on $SO(3)$-bundles with non-trivial second Stiefel-Whitney class, advancing the Atiyah-Floer conjecture.

Contribution

It establishes a framework connecting Lagrangian Floer homology with gauge-theoretic Floer homology in the $SO(3)$ setting, providing foundational results and outlining the proof structure.

Findings

Relation between Lagrangian and gauge-theoretic Floer homologies established

Main results and proof components described for the $SO(3)$-bundle case

Partial analytic details from previous work summarized

Abstract

In this paper the author discuss the relation between Lagrangian Floer homology and Gauge-theory (Donaldson theory) Floer homology. It can be regarded as a version of Atiyah-Floer type conjecture in the case of $SO(3)$-bundle with non-trivial second Stiefel-Whitney class. This is a first of a series of papers, where we describe the main results and geometric and algebraic parts of the proof. The half of analytic detail was in [Fu5] which was published in 1998. The other half will appear in subsequent papers.