Loading...
Please wait, while we are loading the content...
Similar Documents
Getting Acquainted with Intersection Forms
| Content Provider | CiteSeerX |
|---|---|
| Abstract | W E define the intersection form of a 4-manifold, which governs inter-sections of surfaces inside the manifold. We start by representing ev-ery homology 2-class by an embedded surface, then, in section 3.2 (page 115), we explore the properties of the intersection form. Among them is unimodularity, which is essentially equivalent to Poincare duality. An im-portant invariant of an intersection form is its signature, and we discuss how its vanishing is equivalent to the 4-manifold being a boundary of a 5-manifold. After listing a few simple examples of 4-manifolds and their intersection form, in section 3.3 (page 127) we present in some detail the important example of the K3 manifold. Given any closed oriented 4-manifold M, its intersection form is the sym-metric 2-form defined as follows: QM: H2(M;Z) x H2(M;Z) ~ Z QM(a:, {3) = (a: U,B)[MJ. This form is bilinear1 and is represented by a matrix of determinant ± 1. While over 1R this is a recipe for boredom, since this intersection form is defined over the integers (and thus changes of coordinates must be made only through integer-valued matrices), our QM is a quite far-from-trivial object. 1. Notice that QM vanishes on any torsion element, and thus can be thought of as defined on the free part of H2 (M; Z); since our manifolds are assumed simply-connected, torsion is not an issue. |
| File Format | |
| Access Restriction | Open |
| Content Type | Text |
