A characterization of the \(\mathbb Z^n \oplus\mathbb Z(\delta)\) lattice and definite nonunimodular intersection forms (Q2903086)

The intersection form is one of the basic homotopy invariants of a \(4\)-manifold. When the manifold is closed, or has rational homology sphere boundary components, this form is non-degenerate. In this case the determinant of the form is equal to the size of the first homology group. Such a form is a bilinear pairing on a free abelain group. The dual lattice is the set of vectors in the dual space that pair with elements of the lattice to give an integer. There is a pairing on the dual lattice that takes rational values. The first result in this paper establishes the existence of characteristic co-vectors in the dual lattices with small square, in the case of positive-definite forms. When there is a covector attaining the best bound, the lattice is a direct sum of copies of \(\mathbb{Z}\) and one copy of \(\text{det }\mathbb{Z}\). This is a generalization of a theorem of Elkies. The second theorem is an inequality relating the Heegaard-Floer \(d\) invariant of a negative-definite \(4\)-manifold to the determinant of the intersection form. In the last section, Owens and Strle use the results to derive restrictions on the surgeries on torus knots that bound negative definite \(4\)-manifolds. A later paper of the authors that builds on this one, completely characterizes these surgeries [Sel. Math., New Ser. 18, No. 4, 839--854 (2012; Zbl 1268.57006)].