Intersection form

Definition

The intersection form is a bilinear form defined on the middle-dimensional homology of a manifold that measures how submanifolds intersect within that manifold. It plays a crucial role in cobordism theory by providing a way to understand how different manifolds relate to each other through their boundaries and intersections, ultimately influencing the classification of manifolds up to cobordism.

5 Must Know Facts For Your Next Test

1. The intersection form is typically represented as a matrix whose entries reflect the algebraic intersection numbers of submanifolds within a given manifold.
2. For even-dimensional manifolds, the intersection form can be non-degenerate, leading to important implications for the topology and classification of these manifolds.
3. In cobordism theory, the intersection form helps determine whether two manifolds are cobordant by analyzing how their boundaries interact under intersection.
4. The intersection form can reveal information about the signature of a manifold, which is an invariant that counts the difference between the number of positive and negative eigenvalues of the form.
5. When considering oriented manifolds, the intersection form respects orientation and thus can be used to define duality relationships between homology and cohomology groups.

Review Questions

• How does the intersection form contribute to understanding relationships between different manifolds?
  • The intersection form provides a mathematical framework to measure and analyze how different submanifolds intersect within a given manifold. By examining these intersections, one can infer relationships between manifolds, such as whether they are cobordant. This bilinear form captures essential geometric and topological information that can help classify manifolds based on their boundaries and intersection behavior.
• Discuss the significance of non-degeneracy in the context of the intersection form for even-dimensional manifolds.
  • Non-degeneracy of the intersection form in even-dimensional manifolds indicates that the bilinear form has no zero eigenvalues, which suggests that there is a rich structure underlying the manifold's topology. This property allows for stronger conclusions about cobordism classes and can lead to results such as a manifold being classified as spin or non-spin. The implications are significant in distinguishing between different types of manifolds and understanding their geometric properties.
• Evaluate how the intersection form interacts with homology and cohomology in relation to cobordism theory.
  • The intersection form plays a critical role in connecting homology and cohomology theories, particularly in the context of cobordism theory. It acts as an invariant that helps differentiate between various classes of manifolds by reflecting how boundaries intersect in higher dimensions. By analyzing the interaction of this form with homology classes, one can derive powerful results regarding duality and invariance under cobordism, thus providing deeper insights into the nature and classification of manifolds.

Related terms

cobordism: A relation between two manifolds where they can be considered as the boundaries of some higher-dimensional manifold.

homology: A mathematical concept that studies topological spaces by associating sequences of abelian groups or modules to them, representing their multi-dimensional holes.

manifold: A topological space that locally resembles Euclidean space, allowing for the study of shapes and geometries in higher dimensions.