Major Fiber Bundles to Know for Elementary Algebraic Topology

Major fiber bundles are essential structures in algebraic topology, connecting base spaces and fibers in various ways. They help us understand complex topological properties, from simple trivial bundles to intricate examples like the Möbius strip and Hopf fibration.

1. Trivial bundle

   • A trivial bundle is a fiber bundle that is globally product-like, meaning it can be expressed as a product of the base space and the fiber.
   • It is represented as ( B \times F ), where ( B ) is the base space and ( F ) is the fiber.
   • Trivial bundles serve as a foundational example for understanding more complex bundles and their properties.

2. Möbius strip

   • The Möbius strip is a non-orientable surface that can be viewed as a fiber bundle with a circle as the base and a line segment as the fiber.
   • It illustrates the concept of twisting in fiber bundles, as it has a single edge and a single side.
   • The Möbius strip is a classic example of a non-trivial bundle, highlighting the difference between trivial and non-trivial bundles.

3. Tangent bundle

   • The tangent bundle of a manifold consists of all tangent vectors at every point in the manifold, forming a new manifold.
   • It is a vector bundle where the fibers are vector spaces corresponding to the tangent space at each point.
   • The tangent bundle is essential for studying differential geometry and the behavior of curves and surfaces.

4. Hopf fibration

   • The Hopf fibration is a specific example of a fiber bundle where the 3-sphere ( S^3 ) is fibered over the 2-sphere ( S^2 ) with fibers being circles ( S^1 ).
   • It demonstrates a non-trivial topology and is a key example in the study of complex projective spaces.
   • The Hopf fibration has applications in quantum mechanics and the study of fiber bundles in higher dimensions.

5. Principal bundle

   • A principal bundle is a type of fiber bundle where the fibers are groups, specifically Lie groups, acting freely and transitively on the fibers.
   • It is characterized by a structure group that defines the symmetries of the bundle.
   • Principal bundles are fundamental in gauge theory and the study of connections and curvature.

6. Vector bundle

   • A vector bundle is a fiber bundle where the fibers are vector spaces, allowing for linear combinations of vectors at each point in the base space.
   • It is crucial for understanding various mathematical structures, including differential forms and sections.
   • Vector bundles are used extensively in physics, particularly in the formulation of gauge theories.

7. Frame bundle

   • The frame bundle is a specific type of vector bundle that consists of all possible bases (frames) for the tangent spaces of a manifold.
   • It provides a way to study the manifold's geometry and topology by examining the linear structures at each point.
   • The frame bundle is instrumental in defining connections and curvature on manifolds.

8. Sphere bundle

   • A sphere bundle is a fiber bundle where the fibers are spheres, typically denoted as ( S^n ), over a base space.
   • It is used to study the topology of manifolds and their higher-dimensional analogs.
   • Sphere bundles are important in algebraic topology, particularly in the classification of vector bundles.

9. Tautological line bundle

   • The tautological line bundle is a specific line bundle over the projective space, where each fiber corresponds to a line in the vector space.
   • It serves as a fundamental example in the study of line bundles and their properties.
   • The tautological line bundle is crucial for understanding the relationship between geometry and algebraic topology.

10. Stiefel manifold

    • The Stiefel manifold is a space of ordered orthonormal frames in Euclidean space, serving as a principal bundle over the Grassmannian manifold.
    • It plays a significant role in the study of vector bundles and the topology of manifolds.
    • Stiefel manifolds are used in various applications, including robotics and computer vision, to describe configurations of systems.