The Exactness Theorem for Floer Homology

In this article, we study a long exact sequence relating the mstanton homology of two homology 3-spheres which are obtained from each other by + 1 -surgery. We prove that the long exact sequence, via a short exact sequence of chain complex, is the same as the sequence defined by the exact triangle of cobordisms introduced by Floer. One development which has attracted a great deal of attention has been Floer's work on "instanton homology" of 3-manifolds [10], [14], [15], [19], [20], The basic idea is to find Floer homology groups HF*(Y) associated to an oriented 3-manifold Y by studying instantons on the tube Y X R. The theory applies in the first instance to homology 3-sphere Y. A fundamental question in Floer theory is the calculation of the Floer homology groups. A great step forward here was made by Floer who found an "exact triangle" of homomorphisms connecting the Floer homology groups of the 3-manifold Y with those of the 3-manifolds Y', Y" obtained from Y by Dehn surgery on a knot [6], [16], [17], [18]. This paper is based on the works of Floer, Braam and Donaldson. We consider a long exact sequence relating the instanton homology of two homology 3-spheres which are obtained from each other by ±1 -surgery. The third term is a Z^graded homology of the homology S X S which is associated to a knot in the homology 3~sphere via 0-surgery. We prove that the long exact sequence obtained via a short exact sequence of chain complex, is the same as the sequence defined by the exact triangle of cobordisms introduced by Floer in [16], [17]. Suppose that, as above, Y is an oriented homology 3-sphere and X^Y is a Communicated by K. Saito, October 16, 1995. 1991 Mathematics Subject Classification(s): 57N10, 55N35 *Department of Mathematics, National University of Singapore, Singapore 119260