Niemeier lattices, smooth 4-manifolds and instantons
type: Item
instance of: scholarly article
title: Niemeier lattices, smooth 4-manifolds and instantons
zbMATH Open document ID: 1458.57035
author: Christopher Scaduto
published in: Mathematische Annalen
publication date: 2021-02-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1808.10321
review text: Let \(X\) be a smooth, compact, oriented 4-manifold with boundary an integral homology sphere \(Y\). Then \(L_X=H_2(X,{\mathbb Z})/Tor\) is a unimodular lattice. A breaktrough result [\textit{S. K. Donaldson}, J. Differ. Geom. 24, 275--341 (1986; Zbl 0635.57007)] says that if \(Y\) is empty, and \(L_X\) is positive definite, then \(L_X=\langle 1\rangle^n\). For an integral homology 3-sphere \(Y\), let \(E_i(Y)\) be the set of isomorphism classes of minimal definite unimodular lattices \(L\) of rank \(i\) such that there exists an \(X\) as above with boundary \(Y\) and \(L_X=L \oplus \langle 1\rangle^n\). The rank of an even positive definite unimodular lattice is a multiple of 8. There is only one lattice of rank 8, two of rank 16, and 24 of rank 24. This paper focuses on lattices of rank 24, which are called Niemeier lattices. The main result is that there are different \(Y\) such that \(E_{24}(Y)\) are non-empty and distinct sets. Therefore, the lattices that can arise for boundaries of smooth 4-manifolds can depend on finer information than just the ranks of the lattices. The 3-manifolds which are taken are +1 Dehn surgery \(Y=S_ 1^3(K)\) on torus knots \(K=T_{n,m} \subset S^3\). The main tool are the relations in the instanton cohomology ring of a genus \(g\) surface \(\Sigma\) times a circle, computed in [\textit{V. Muñoz}, Topology 38, No. 3, 517--528 (1999; Zbl 0927.57032)], with mod 2, mod 4 and mod 8 coefficients. For a lattice \(L\), one defines the 4-genus \(g_4(L)\) as the minimum \(g\geq 0\) such that there exists a smooth, closed, oriented 4-manifold \(W\) with \(b_2^-(W)=1\) and \(\Sigma\subset W\) of genus \(g\) and \(\Sigma^2=-1\), and \(L_W=\langle \Sigma \rangle \oplus L \oplus \langle 1\rangle^n\). For a knot \(K\subset S^3\), one has \(g_4(K)\geq g_4(L)\) if \(L\in E_{24}(S_1^3(K))\). The technical part of the paper consists in computing (or at least bounding) the numbers \(g_4(L)\) for many Niemeier lattices \(L\). The upper bounds come from ge…
reviewed by: Vicente Muñoz
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57K40
zbMATH DE Number: 7307518
zbMATH Keywords: integral homology sphere
zbMATH Keywords: 4-manifold
zbMATH Keywords: unimodular lattice
zbMATH Keywords: instanton Floer homology
zbMATH Keywords: Niemeier lattice
MaRDI profile type: Publication
OpenAlex ID: W3091828707
arXiv ID: 1808.10321
cites work: Heegaard Floer correction terms, with a twist
cites work: Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices
cites work: Q4219187
cites work: Connections, cohomology and the intersection forms of 4-manifolds
cites work: The orientation of Yang-Mills moduli spaces and 4-manifold topology
cites work: A characterization of the \(\mathbb Z^ n\) lattice
cites work: The Seiberg-Witten equations and four-manifolds with boundary
cites work: Equivariant aspects of Yang-Mills Floer theory
cites work: An inequality for the \(h\)-invariant in instanton Floer theory.
cites work: Monopole Floer homology for rational homology 3-spheres
cites work: On a class of rational cuspidal plane curves
cites work: On definite lattices bounded by integer surgeries along knots with slice genus at most 2
cites work: Q4700561
cites work: Ring structure of the Floer cohomology of \(\Sigma\times S^1\)
cites work: Q3220682
cites work: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary
cites work: Knot Floer homology and the four-ball genus
cites work: Lens space surgeries and a conjecture of Goda and Teragaito
DOI: 10.1007/S00208-020-02060-Y
description: scientific article