Lattice Cohomology

Némethi, András and Ágoston, Tamás (2024) Lattice Cohomology. In: Singularities and Low Dimensional Topology. Bolyai Society Mathematical Studies (30). Springer Nature Switzerland, Cham, pp. 107-138. ISBN 9783031566103; 9783031566110; 9783031566134

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Abstract

This note is a gentle introduction to the lattice cohomology of isolated complex analytic germs. The analytic version is sensitive to the analytic type of the germs. It can be defined in any positive dimension, in particular for reduced curve singularities as well. Here we will treat mostly the curve case. The topological version is defined for the topological type of an isolated surface singularity with a rational homology sphere link. These cohomology theories are categorifications of famous numerical invariants. E.g., the Euler characteristic of the lattice cohomology of a reduced curve singularity is its delta invariant. In section 1 we review some notations and elementary properties of singular analytic germs. In the case of isolated plane curve singularities we compare numerical invariants read from the embedded topological type with invariants read from the abstract analytic type. In section 2 we treat the lattice cohomology. We provide some combinatorial statements and also several examples both in the curve and surface cases.

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