Lattices and correction terms

2018/07/13 by Kyle P. Larson, Kyle Larson, Larson, Kyle
Computer Science · Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT

paper · pdf · doi:10.48550/arxiv.1807.05098

9 pages. Comments welcome. Second version contains minor improvements, and a new example

arxiv created 2019/02/22 · arxiv updated 2019/02/25

Abstract

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction for a rational homology 3-sphere to bound a rational homology 4-ball coming from Donaldson's theorem on definite intersection forms of 4-manifolds. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.

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