Seiberg-Witten theory on 4-manifolds with periodic ends

Abstract

In this thesis we prove analytic results about a cohomotopical Seiberg-Witten theory for a Riemannian, Spin$^c$(4), 4-manifold with periodic ends, $(X, g, {\tau})$ . Our results show that, under certain technical assumptions on $(X, g, {\tau})$, this new version is coherent and leads to Seiberg-Witten type invariants for this new class of 4-manifolds. In the first part, using Taubes criteria for end-periodic operators, we show that for a Riemannian 4-manifold with periodic ends, $(X, g)$, verifying certain topological conditions, the Laplacian, $\Delta_+ : L^2_2({\Lambda^2}_+) \rightarrow L^2({\Lambda^2}_+)$, is a Fredholm operator. This allows us to prove a Hodge type decomposition for positively weighted Sobolev 1-forms on $(X,g)$. We also prove, assuming non-negative scalar curvature on each end and certain technical topological conditions, that the associated Dirac operator associated with an end-periodic connection (which is ASD at infinity) is Fredholm. In the second part we establish an isomorphism between the de Rham cohomology group, $H^{1}_{\mathrm{dR}}(X,i\mathbb{R})$ (which is a topological invariant of X) and the harmonic group intervening in the above Hodge type decomposition of the space of positively weighted 1-forms on $(X,g)$. We also prove two short exact sequences relating the gauge group of the Seiberg-Witten moduli problem and the cohomology group $H^{1}(X, 2{\pi}i\mathbb{Z})$. In the third part, we prove the main results: the coercivity of the Seiberg-Witten map and the compactness of the moduli space for a 4-manifold with periodic ends, $(X,g,{\tau})$, verifying the above conditions. Finally, using the coercitivity property, we show that a Seiberg-Witten type cohomotopy invariant associated to $(X, g, {\tau})$ can be defined