Three-Manifolds and Manifolds with Cylindrical Ends

Contents Introduction. 1 Chapter 1. Four-dimensional Theory. 4 1. Setup. 4 2. A slice theorem for singular connections. 9 3. Laplacians on forms. 17 Chapter 2. Three-dimensional Theory. 22 1. Setup. 22 2. The Chern-Simons functional. 24 3. The deformation complex. 27 Chapter 3. Singular connections on manifolds with cylindrical ends. 33 1. Setup. 33 2. The translation-invariant case. 35 3. Global theory. 40 4. Some index formulae. 42 Loose ends and speculations. 47 Appendix A. Elliptic edge operators in a nutshell. 49 Bibliography 56 ii Introduction. This thesis can be seen as a first step towards defining a Floer homology for the singular connections studied by Kronheimer and Mrowka in their work on the structure of Donaldson's polynomial invariants [KM]. By investigating certain spaces of connections on a homology three-sphere which are singular along a knot, such a Floer homology would provide possibly a new invariant for the knot. We will first briefly recall the key ideas of the classical instanton homology defined by Floer [F1] for a homology three–sphere and describe the singular connections considered by Kronheimer and Mrowka. We will then present the content of the thesis. Floer homology is obtained by applying an infinite–dimensional version of Morse theory to the Chern-Simons functional defined over the space of connections in a principal bundle (with e.g. structure group SU(2)) over a closed, oriented, three–manifold M. The set R of critical points of this functional are the flat connections and the gradient-flow lines correspond to anti–self–dual connections over the cylinder M ×R. After generic perturbation, the critical points form a finite set of isolated connections. The index of the critical points is always infinite, but it is nevertheless possible to define a relative index between two critical points by the spectral flow [APS] of the Hessian of the Chern-Simons functional. After dividing by the action of the gauge group, this spectral flow is well-defined modulo 8. It follows that the free abelian group R generated by the set of critical points is Z/8–graded. If we consider two critical points α and β with consecutive gradings, the space of flow lines starting from α and arriving at β is generically a zero–dimensional, oriented, compact manifold, i.e. a finite set of signed points. Counting those flow lines as in classical finite–dimensional Morse theory yields a boundary map δ : R → R, and Floer proved that δ 2 = 0 by …