Seiberg-Witten Equations on Tubes

In [10] we began a study of the 3-dimensional Seiberg-Witten equations on Seifert manifolds with two goals in mind: to compute the Seiberg-Witten-Floer (SWF) homology of these manifolds and ultimately, to produce gluing formulæ for the 4dimensional Seiberg-Witten invariants. A first difficulty one must overcome has to do with the less than obvious nature of the solutions of the 3-dimensional Seiberg-Witten equations. We dealt with this issue in [10] where we studied the behavior of these solutions as the Seifert fibration collapses onto its base (i.e. the background metric is shrunk in the fiber direction). As the metric is deformed, the solutions of the SW equations converge to solutions of some adiabatic Seiberg-Witten equations. These are variational equations and can be solved explicitly. Moreover, these adiabatic equations are very simple zeroth order perturbations of the original ones which suggests that the Morse theory for the adiabatic equations produces the same results as the original ones (which may have to be perturbed anyway to be placed in a generic framework). A key fact established in [10] was that, in the case of a smooth S1-bundle N over a Riemann surface Σ equipped with a product-like metric with sufficiently short fibers, the adiabatic Morse function is Bott nondegenerate along the irreducible part of its critical set. This fact makes the adiabatic theory even more tempting to use for Floer theory computations. The Bott extension of Morse theory (in the form described for example in [1]) describes a spectral sequence associated to a Morse-Bott function converging to the cohomology of the background manifold. This approach can be extended to our infinite dimensional situation as in [4] for the instanton homology. The (co)boundary operators of this spectral sequence are defined in terms of the tunnelings between different components of the critical set, i.e. connecting trajectories of the gradient flow. In the case at hand, the gradient flow equations are