On symplectic 4-manifolds and contact 5-manifolds [Elektronische Ressource] / Mark Hamilton

Mark HamiltonOn symplectic 4 manifoldsand contact 5 manifoldsDissertation an der Fakultat¨ fur¨Mathematik, Informatik und Statistik derLudwig Maximilians Universit at¨ Munchen¨6. Mai 2008Erster Gutachter: Prof. Dieter Kotschick, D. Phil. (Oxon)(Ludwig Maximilians Uni versitat¨ Munchen)¨Zweiter Gutachter: Prof. Hansjor¨ g Geiges, Ph. D. (Cantab)(Universitat¨ zu Koln)¨Tag der mundlichen¨ Prufung:¨ 27. Juni 2008AbstractIn this thesis we prove some results on symplectic structures on 4 dimensional manifolds and contactstructures on 5 dimensional manifolds. We begin by discussing the relation between holomorphic andsymplectic minimality for Kahler¨ surfaces and the irreducibility of minimal simply connected symplec tic 4 manifolds under connected sum. We also prove a result on the conformal systoles of symplectic4 manifolds. For the generalized fibre sum construction of 4 manifolds we calculate the integral homol ogy groups if the summation is along embedded surfaces with trivial normal bundle. In the symplecticcase we derive a formula for the canonical class of the generalized fibre sum and give several appli cations, in particular to the geography of simply connected symplectic 4 manifolds whose canonicalclass is divisible by a given integer. We also use branched coverings of complex surfaces of generaltype to construct simply connected algebraic surfaces with divisible canonical class.