Niemeier lattices, smooth 4-manifolds and instantons
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Mathematische Annalen
Niemeier lattices, smooth 4-manifolds and instantons Christopher Scaduto1 Received: 11 March 2019 / Revised: 28 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that the set of even positive definite lattices that arise from smooth, simplyconnected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.
1 Introduction Let X be a smooth, compact, oriented 4-manifold. The intersection form of X is the free abelian group L X = H2 (X ; Z)/Tor equipped with the symmetric bilinear form L X ⊗ L X → Z defined by the intersection of 2-cycles. A well-known result of Donaldson [4,5] says that if X has no boundary, and L X is positive definite, then it is equivalent over the integers to a diagonal lattice 1n . In general, a lattice is a free abelian group L of finite rank equipped with a symmetric bilinear form. We write x · y for the pairing of x, y ∈ L. A lattice L is unimodular if it has a basis {ei } for which | det(ei · e j )| = 1. If X as above has an integer homology 3-sphere boundary Y , then L X is a unimodular lattice. A lattice L is even if x · x is an even integer for every x ∈ L. A definite lattice is minimal if it has no vectors of absolute norm 1. Fix an integer homology 3-sphere Y . Write L (Y ) for the set of isomorphism classes of minimal definite unimodular lattices L such that there exists a smooth, compact, oriented 4-manifold X without 2-torsion in H∗ (X ; Z), ∂ X = Y , and L X = L ⊕ 1n for some n ≥ 0. Write Ei (Y ) ⊂ L (Y ) for the subset of even rank i lattices. That Ei (Y )
Communicated by Thomas Schick. Christopher Scaduto: the author was supported by NSF grant DMS-1503100.
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Christopher Scaduto [email protected] Simons Center for Geometry and Physics, New York, USA
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is empty for i large enough was proven by Frøyshov using Seiberg-Witten theory [10], and also follows from Heegaard Floer theory [16]. The rank of an even positive definite unimodular lattice is an integral multiple of 8. There is only one such lattice of rank 8, and two of rank 16. In rank 24, there are 24 such lattices, the Niemeier lattices, see e.g. [3, Ch.6], the set of which we denote by N24 . The number beyond rank 24 grows quickly, and those of rank 32 already make up more than a billion. Write S13 (K ) for +1 Dehn surgery on a knot K ⊂ S 3 , and Tn,m for the positive (n, m) torus knot. Theorem 1.1 E24 (S13 (T2,11 )) and E24 (S13 (T4,5 )) are distinct and nonempty subsets of N24 . This result contrasts with the constraints on E24 (Y ) imposed by Seiberg-Witten and Heegaard Floer theory. In those contexts, only the ranks of even definite lattices bounded by a homology 3-sphere Y are constrained, and thus either E24 (Y ) is empty or no further information is obtained. The knots T2,11 and T4,5 in Theorem 1.1 may be replaced by other knots; see Sect.
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