Niemeier lattices, smooth 4-manifolds and instantons
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 geometric constructions of a suitable \(W\). The lower bounds come from the relations in the instanton cohomology ring of \(\Sigma\times S^1\).
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