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

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.