Nearly Unimodular Quadratic Forms

Abstract

Quadratic forms on lattices (free modules of finite rank) over Z have received wide attention in recent years because of their interactions with group theory, coding theory, and topology; lattices with unimodular quadratic forms have been a subject of special interest in all these areas. For example, unimodular lattices are important to topologists in connection with the classification of manifolds: If M is a closed, oriented, simply connected 4-manifold, then the homology group H2(M; Z) is a unimodular 2-lattice with respect to the quadratic form induced by intersection numbers, and there is an orientationpreserving homotopy equivalence of two such manifolds if and only if the corresponding lattices are isometric (see [M-H, p. 103], or [Ki, p. 22]). Freedman ([F-Q, Sec. 10.1]) has shown that every unimodular 2-lattice occurs as a homology group of a simply connected, closed, topological 4-manifold in this way. If the form is even (type II) then the isometry class of the lattice determines the manifold uniquely; if the form is odd (type I) there are exactly two associated manifolds (and they are distinguished by their Kirby-Siebenmann invariants). Two indefinite unimodular 2-lattices are isometric if and only if they have the same rank, type, and signature (see [M-H, Chap. 2], or [S, Chap. 5]); but classification in the definite case remains a major open problem. In low dimensions the classes of definite unimodular lattices are known: the work of Kneser [Kn], Niemeier [Ni], and Borcherds [B] has produced an explicit listing of the classes of rank < 25. (Those of rank < 24 appear in Chapters 16 and 17 of [C-S].) For example, there are only 24 classes of even unimodular lattices of rank 24. But mass formula considerations show that there are more than 107 such classes of rank 32 and more than 1051 such classes in rank 40, suggesting that a complete set of computable invariants may be extraordinarily difficult to find. Recently Newman [Ne2] has developed a matrix algorithm that yields a surprisingly simple reduced form for unimodular lattices. We will look at his result in detail, but the upshot is that every unimodular lattice has a primitive sublattice of codimension 1 belonging to a rather special family, which we call