Introduction: This paper is an exposition of A. Floer’s work which was completed circa 1989 and distributed in the shape of the two preprints [ibid., 77-97 (1995; Zbl 0996.57503), see above; Instanton homology for knots (preprint)]. A description of the results was published in the Durham Proceedings [A. Floer, Lond. Math. Soc. Lect. Note Ser. 150, 97-114 (1990; Zbl 0788.57008)]. In this first part of the paper we deal with the “gauge theory” content of this work of Floer: that is, the proofs of the exact triangle in the first preprint [loc. cit.] and the “excision axiom” of the second preprint [loc. cit.]. This part is written with the aim of coming quickly to grips with the main geometrical points involved. The second part of our paper takes the topics further, introducing a more general framework for the results and describing the calculation scheme Floer developed in the second preprint [loc. cit.]. The salient points here are in Sections II.1.3 where automorphisms of Floer homology are explained and in II.1.4, where Floer homology is computed for some important special manifolds. In section II.2.2 the exact triangle is discussed in a general setting, and in II.3.1, where it is explained how Kirby calculus gives rise to a set of exact sequences.
For the entire collection see [Zbl 0824.00019].