Intersection form
Introduction
The intersection form is a fundamental concept in the field of algebraic topology and differential geometry, particularly in the study of four-manifolds. It is a bilinear form that provides crucial insights into the topology of manifolds by encoding information about how submanifolds intersect within a given manifold. The intersection form is particularly significant in the classification of smooth structures on four-dimensional manifolds, playing a pivotal role in Donaldson's theorem and the study of exotic smooth structures.
Definition and Basic Properties
The intersection form is defined on a smooth, compact, oriented manifold \( M \) of dimension \( 4k \). It is a bilinear form on the middle-dimensional homology group \( H_{2k}(M; \mathbb{Z}) \). Given two homology classes \( [a], [b] \in H_{2k}(M; \mathbb{Z}) \), their intersection number is denoted by \( [a] \cdot [b] \) and is defined as the algebraic count of intersection points of representative cycles of these classes. The intersection form is symmetric, meaning \( [a] \cdot [b] = [b] \cdot [a] \).
In the case of four-manifolds, the intersection form is a quadratic form on \( H_2(M; \mathbb{Z}) \). For a simply connected four-manifold, this form is unimodular, meaning its determinant is \( \pm 1 \). The signature of the intersection form, which is the difference between the number of positive and negative eigenvalues, is an important topological invariant.
Algebraic Topology Context
In algebraic topology, the intersection form is a tool for understanding the structure of manifolds. It is closely related to the Poincaré duality, which provides an isomorphism between the homology and cohomology groups of a manifold. The intersection form can be seen as a manifestation of this duality, as it pairs homology classes in a way that reflects the manifold's topology.
The intersection form also plays a role in the Hodge theory, where it is related to the Hodge star operator and the decomposition of differential forms. In this context, the intersection form can be used to study the harmonic forms on a manifold, providing insights into its geometric structure.
Applications in Four-Manifolds
The study of four-manifolds is a rich and complex area of mathematics, and the intersection form is a central tool in this field. One of the most significant results involving the intersection form is Donaldson's theorem, which classifies smooth structures on simply connected four-manifolds. Donaldson showed that the intersection form of a smooth, simply connected four-manifold is diagonalizable over the integers, a result that has profound implications for the study of smooth structures.
The intersection form is also crucial in the study of exotic smooth structures, which are smooth structures on topological manifolds that are not diffeomorphic to the standard smooth structure. The existence of exotic smooth structures on four-manifolds is a striking phenomenon, and the intersection form provides a way to distinguish between different smooth structures.
Intersection Form and Gauge Theory
Gauge theory, particularly the study of Yang-Mills theory, has deep connections with the intersection form. In the context of four-manifolds, gauge theory provides a framework for studying the moduli spaces of solutions to the Yang-Mills equations. These moduli spaces are often smooth manifolds themselves, and their topology is intimately related to the intersection form of the underlying four-manifold.
The Seiberg-Witten invariants are another important tool in this area, providing a way to study the differential topology of four-manifolds. These invariants are related to the intersection form and provide a means of distinguishing between different smooth structures on a manifold.
Computation of Intersection Forms
The computation of intersection forms is a non-trivial task that often involves sophisticated techniques from algebraic topology and differential geometry. For a given four-manifold, the intersection form can be computed by choosing a basis for the middle-dimensional homology group and calculating the intersection numbers of the basis elements.
In some cases, the intersection form can be determined using Morse theory, which relates the topology of a manifold to the critical points of a smooth function. By studying the critical points and their indices, one can gain insights into the intersection form and other topological invariants of the manifold.
Examples and Case Studies
Several well-known manifolds have intersection forms that are of particular interest. For example, the intersection form of the K3 surface, a complex surface with remarkable geometric properties, is an important object of study in both mathematics and theoretical physics. The K3 surface has an intersection form that is even and unimodular, with signature zero, making it a key example in the study of four-manifolds.
Another important example is the intersection form of the complex projective space \( \mathbb{CP}^2 \), which is a fundamental object in algebraic geometry. The intersection form of \( \mathbb{CP}^2 \) is given by the matrix \((1)\), reflecting its simple topological structure.