Normal bundle

Definition

A normal bundle is a vector bundle that describes the directions in which one can move away from a submanifold within a larger manifold. It captures how the submanifold sits inside the ambient space, providing crucial information about its geometry and topology. The normal bundle plays an important role in various mathematical contexts, including the study of Gysin homomorphisms and push-forward maps as well as the foundational concepts of vector bundles.

5 Must Know Facts For Your Next Test

1. The normal bundle can be thought of as capturing all possible directions orthogonal to the tangent space of the submanifold.
2. If a submanifold is embedded in a manifold, the normal bundle is determined by the ambient space's tangent space at points on the submanifold.
3. The rank of the normal bundle corresponds to the codimension of the submanifold in the ambient manifold.
4. In terms of Gysin homomorphisms, the normal bundle helps define push-forward maps that relate cohomology classes between different manifolds.
5. Understanding the normal bundle is essential for many applications in algebraic topology, differential geometry, and mathematical physics.

Review Questions

• How does the normal bundle relate to the tangent spaces of a submanifold?
  • The normal bundle consists of vectors that are orthogonal to the tangent space of the submanifold at each point. While the tangent space captures directions in which one can move along the submanifold, the normal bundle represents directions in which one can move away from it into the ambient manifold. This distinction is crucial for understanding how the submanifold is embedded within its surrounding space.
• Discuss how Gysin homomorphisms utilize the normal bundle when performing push-forward maps in algebraic topology.
  • Gysin homomorphisms leverage the properties of the normal bundle to establish relationships between cohomology groups of a submanifold and its ambient manifold. When performing push-forward maps, these homomorphisms consider how classes in the cohomology of the submanifold can be represented within the larger manifold by incorporating information from the normal bundle. This interaction allows for more profound insights into how topological features are preserved or transformed under mappings.
• Evaluate the implications of understanding normal bundles in both algebraic topology and differential geometry, highlighting any interconnected concepts.
  • Understanding normal bundles provides critical insight into both algebraic topology and differential geometry as they intersect with concepts like embedding, intersection theory, and cohomology. In differential geometry, knowing how a submanifold sits inside an ambient space reveals geometric properties like curvature and local flatness. In algebraic topology, normal bundles facilitate calculations involving Gysin homomorphisms, which link different topological spaces. Together, these perspectives enrich our understanding of manifold theory and its applications across mathematics.

Related terms

Submanifold: A submanifold is a subset of a manifold that itself has the structure of a manifold, allowing for differential calculus to be applied.

Tangent Bundle: The tangent bundle is a vector bundle that represents all possible tangent vectors at every point of a manifold, reflecting its local geometry.

Vector Bundle:

A vector bundle is a topological construction that associates a vector space to each point of a base space, allowing for the study of fields and sections.