Abstract
Let L be a nonunimodular definite lattice, L ∗ its dual lattice, and λ the discriminant form on L ∗∕L. 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 (L ∗∕L, λ). 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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Acknowledgements
The author is pleased to thank Marco Golla, Joshua Greene, Brendan Owens, and Sašo Strle for helpful conversations and comments. In addition, we thank Marco Golla for sharing Proposition 2.1 with the author, and a referee for helpful suggestions. This research was partially supported by NKFIH Grant K112735.
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Larson, K. (2021). Lattices and Correction Terms. In: Fernández de Bobadilla, J., László, T., Stipsicz, A. (eds) Singularities and Their Interaction with Geometry and Low Dimensional Topology . Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-61958-9_11
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