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One of the main questions in the theory of normal surface singularities is tounderstand the relations between their geometry and topology. The latticecohomology is an important tool in the study of topological properties of aplumbed 3-manifold M associated with a connected negative definite plumbinggraph G. It connects the topological properties with analytic ones when M isrealized as a singularity link, i.e. when G is a good resolution graph of thesingularity. Its computation is based on the (Riemann-Roch) weights of thelattice points of \Z^s, where s is the number of vertices of G. The first partof the thesis reduces the rank of this lattice to the number of `bad' verticesof the graph. Usually, the geometry/topology of M is encoded exactly by these`bad' vertices and their number measures how far the plumbing graph stays froma rational one. In the second part, we identify the following three objects:the Seiberg-Witten invariant of a plumbed 3-manifold, the periodic constant ofits topological Poincare series, and a coefficient of an equivariantmultivariable Ehrhart polynomial. For this, we construct the correspondingpolytope from the plumbing graph, together with an action of H_1(M,\Z), and wedevelop Ehrhart theory for them. The effect of the reduction appears also atthe level of the multivariable topological Poincare series, simplifying thecorresponding polytope and the Ehrhart theory as well. We end the thesis withdetailed calculations and examples.

Lattice cohomology and Seiberg-Witten invariants of normal surface singularities