Floer Homology and Splittings of Manifolds

Abstract

In [F], Floer defined a mod8 graded homology group I,(M) for an oriented integral homology 3-sphere M. It is an invariant of the differentiable structure of M. Roughly speaking I *(M) is a homology group coming from the Morse theory of the Chern-Simons functional f on the infinite-dimensional manifold of all the gauge equivalence classes of irreducible connections on the principal SU(2)-bundle M X SU(2). The critical point set of f is the set of the gauge equivalence classes of irreducible flat connections. In general, these critical sets may be degenerate and f not be a genuine Morse functional; a suitable perturbation of f is needed to define I * (M). In this paper we consider only non-degenerate critical points of f. This is partly because we want to avoid some inessential technical complications and partly because it is sufficient for our computations of I * (Nk) for homology 3-spheres Nk obtained by Dehn surgery along the figure eight knot in S'. For a smooth connection A on M X SU(2), a self-adjoint Fredholm operator DA is defined (Section 2). The gauge equivalence class [A] of an irreducible flat connection A is a non-degenerate critical point of f if and only if Ker DA = 0. In this case [A] determines a generator of the mod 8 graded chain group of l* (M). The mod 8 degree d([A]) is related to the spectral flow invariant (Section 2) as follows. Let AO and A, be two irreducible flat connections on M X SU(2) such that Ker DA. = 0 = Ker DA,. Let (At) be a smooth path of smooth connections on M X SU(2) connecting AO and A1. Then d([AI]) d([A 0]) is the mod 8 reduction of the spectral flow of the path of the self-adjoint Fredholm operators { DA }. The object of this paper is to give a practical method of calculation of the above spectral flow when M is split as M = M1 U M2, where M1 n M2 = aM, = dM2 = X and X is an orientable surface of genus g (> 2). Such a decomposition in 3-dimensional gauge theory was first considered by C. H.