8.1 Fibrations and fiber bundles

Fibrations and fiber bundles are key concepts in algebraic topology. They generalize covering spaces, allowing for more complex fibers and global structures. These tools help us understand how spaces are built from simpler pieces.

The homotopy lifting property of fibrations connects the topology of different spaces. Fiber bundles, with their local product structure, offer insights into global twisting. Both concepts are crucial for studying geometric and topological relationships.

Fibrations and Fiber Bundles

Definition and Key Properties

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• A fibration is a continuous surjective map $p: E \to B$ between topological spaces $E$ and $B$ that satisfies the homotopy lifting property for all spaces
  • The homotopy lifting property states that given a homotopy $f: X \times I \to B$ and a lift $g: X \times \{0\} \to E$ of $f|_{X \times \{0\}}$, there exists a homotopy $\tilde{f}: X \times I \to E$ lifting $f$ such that $\tilde{f}|_{X \times \{0\}} = g$
• A fiber bundle is a continuous surjective map $p: E \to B$ between topological spaces $E$ and $B$, where each point in $B$ has a neighborhood $U$ such that $p^{-1}(U)$ is homeomorphic to the product space $U \times F$ for some fixed topological space $F$ called the fiber
• The total space $E$ of a fiber bundle locally looks like the product of the base space $B$ and the fiber $F$, but globally it may have a different structure due to twisting or nontrivial gluing of the local product structures
• Fibrations and fiber bundles generalize the concept of a covering space, allowing for more flexibility in the fibers and the global structure (Möbius strip, Hopf fibration)

Examples and Constructions

• The projection map $p: E \to B$ from a product space $E = B \times F$ to its first factor $B$ is a trivial fibration and a trivial fiber bundle with fiber $F$
• The Möbius strip is a nontrivial fiber bundle over the circle $S^1$ with fiber the unit interval $[0, 1]$, obtained by gluing the ends of $[0, 1] \times [0, 1]$ with a half-twist
• The Hopf fibration is a nontrivial fibration from the 3-sphere $S^3$ to the 2-sphere $S^2$, with each fiber being a circle $S^1$
  • It can be constructed using the complex numbers and quaternions
• The tangent bundle of a smooth manifold $M$ is a vector bundle over $M$, with each fiber being the tangent space at a point of $M$
• Principal $G$-bundles are fiber bundles with fiber a Lie group $G$, where $G$ acts freely and transitively on each fiber
  • They play a crucial role in the theory of gauge fields and characteristic classes

Constructing Fibrations and Fiber Bundles

Local Trivializations and Transition Functions

• Locally, a fiber bundle looks like a product space $U \times F$, where $U$ is an open set in the base space $B$ and $F$ is the fiber
  • This local trivialization allows for the study of local properties of the bundle
• The global structure of a fiber bundle is determined by the gluing maps between the local trivializations, called transition functions
  • These functions encode how the local product structures are twisted or identified when moving around the base space
• Constructing a fiber bundle involves specifying the local trivializations and the transition functions that glue them together consistently (cocycle condition)

Pullbacks and Associated Bundles

• Given a fibration $p: E \to B$ and a continuous map $f: X \to B$, the pullback of $p$ along $f$ is a fibration $f^*p: f^*E \to X$ over $X$ that captures the idea of "restricting" the fibration $p$ to the space $X$ via the map $f$
  • The pullback is a useful construction for studying the behavior of fibrations under base change and for defining induced bundles
• Associated bundles are fiber bundles constructed from a principal $G$-bundle $P \to B$ and a space $F$ with a $G$-action
  • The associated bundle $P \times_G F$ has fiber $F$ and is obtained by taking the quotient of $P \times F$ by the diagonal $G$-action
  • Many important bundles in geometry and physics, such as vector bundles and spinor bundles, can be constructed as associated bundles to a principal bundle

Structure of Fibrations and Fiber Bundles

Homotopy Lifting Property and Long Exact Sequences

• The global structure of a fibration is captured by the homotopy lifting property, which relates the topology of the base space to the topology of the total space and the fibers
• The homotopy lifting property allows for the construction of lifts of homotopies and the study of the homotopy theory of the spaces involved
• A fibration $p: E \to B$ with fiber $F$ gives rise to a long exact sequence of homotopy groups: $\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots$
  • This sequence relates the homotopy groups of the total space, base space, and fiber, and provides a powerful tool for computing and understanding their relationships

Characteristic Classes and Obstructions

• The global structure of a fiber bundle can be classified by its characteristic classes, which are cohomology classes that measure the nontriviality of the bundle
  • Characteristic classes provide obstructions to the existence of certain cross-sections or lifts (Euler class, Chern classes, Pontryagin classes)
• The vanishing of certain characteristic classes is often a necessary condition for a bundle to admit a specific structure or property (parallelizability, existence of a connection with prescribed curvature)
• Characteristic classes can be used to distinguish between different bundles over the same base space and to study the topology of the base space itself (Chern character, Stiefel-Whitney classes)

Fibrations vs Other Topological Structures

Comparison with Covering Spaces and Vector Bundles

• Fibrations and fiber bundles generalize the concept of a covering space, allowing for more general fibers and a more flexible global structure
  • Covering spaces are fibrations and fiber bundles with discrete fibers
• Vector bundles are a special case of fiber bundles where the fibers are vector spaces, and the transition functions are linear maps
  • They play a fundamental role in differential geometry and topology (tangent bundle, normal bundle)
• The theory of fibrations and fiber bundles provides a unifying framework for studying covering spaces, vector bundles, and other geometric structures

Sheaves and Local-to-Global Principles

• Sheaves are another generalization of fiber bundles, where the fibers are allowed to vary over the base space and are equipped with a suitable topology and algebraic structure
  • Sheaves provide a framework for studying local-to-global properties in geometry and analysis (sheaf cohomology, sheaf of sections)
• The local-to-global principles embodied by sheaves and fibrations allow for the study of global properties of a space by understanding its local behavior and how it is glued together
• Sheaf theory and the theory of fibrations and fiber bundles provide powerful tools for understanding the interplay between local and global phenomena in mathematics (Čech cohomology, Leray spectral sequence)