An inequality for the \(h\)-invariant in instanton Floer theory.
In [Topology 41, 525--552 (2002; Zbl 0999.57032)], the author defines a non-trivial homomorphism \(h\) from the homology cobordism group of oriented integral homology \(3\)-spheres to the integers, using instanton Floer theory. In this paper some new properties of this invariant are obtained. Let \(X\) be a smooth, compact, oriented \(4\)-manifold with a homology sphere \(Y\) as boundary and \(b_2^+(X)=1\). Let \(\Sigma\subset X\) be a closed oriented embedded surface of genus \(g\) and self-intersection \(\Sigma\cdot\Sigma =1\). Let \(L\subset H^2(X;\mathbb{Z})/\text{torsion}\) be the sublattice annihilating \([\Sigma]\). Then the main result of the paper is the following formula: \[ h(Y)+\lceil g/2\rceil \geq e(L), \] where \(\lceil x \rceil\) denotes the smallest integer \(\geq x\) and \(e(L)\) is an invariant of the definite lattice \(L\) defined as follows: First \(w\in L\) is extremal if \(| w^2|\leq| z^2| \) for any \(z\in w + 2L\). Then \(e(L)\) is defined as the supremum of the quantities \(\lceil | w^2| -m \rceil/4\), where \(w\) is extremal, \(m\geq 0\), \(w^2\equiv m\pmod 2\) and \[ \eta(L,w,a,m)=\sum_{z\in w+2L} (-1)^{((z+w)/2)^2} (a\cdot z)^m \neq 0, \] for some \(a\in\Hom(L,\mathbb{Z})\). Note that \(e(L)\) has nice properties: \(e(L) \geq 0\), \(e(L)=0\) if \(L\) is diagonal, \(e(-k E_8\oplus L)=k\) if \(L\) is diagonal. The argument of proof consists on considering the blow-up \(X\# \overline{\mathbb C P}^2\) and the surfaces \(\Sigma^{\pm} = \Sigma \# (\pm E)\), of self-intersection zero, where \(E\) is the exceptional divisor. Then consider the moduli space of ASD connections for the family of metrics which stretch the metric along \(\Sigma^{\pm}\) for \(t\to \pm \infty\). Cut it down with a cycle of the form \(x^{n-2q} (x_1^2-4)^q\) until it is \(1\)-dimensional. Here \(x\) and \(x_1\) are cycles of the moduli space associated under the \(\mu\)-map to a fixed point in \(X\) well away from \(\Sigma^{\pm}\) and a point close to \(\Sigma^{\pm}\). The result follows by counting the number of ends of the resulting \(1\)-dimensional space, together with the following two facts: the reducible connections give rise to the number \(\eta(L,w,a,m)\), whilst \((x_1^2-4)^q\) is zero on Floer homology \(HF(\Sigma^{\pm} \times S^1)/\mu(H_1(\Sigma))\) for \(q=\lceil g/2 \rceil\), by a result in [\textit{V. Muñoz}, Topology 38, 517--528 (1999; Zbl 0927.57032)]. Some nice features are worth pointing out. First, the case \(Y=S^3\) recovers the result in [\textit{S. K. Donaldson}, J. Differ. Geom. 18, 279--315 (1983; Zbl 0507.57010)]. Second, the main result in this paper allows to compute \(h(Y)\) for some examples, e.g., \(h(\Sigma(2,2k-1,4k-3)) = \lceil (k+1)/2\rceil\), \(k\geq 2\).
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