Question

Show that \(\phi_{u}: \pi^{-1}(U) \rightarrow U \times F\) defined by \(\phi_{u}([p, \xi \rrbracket)=(\pi(p)\), \(\left.s_{u}(p) \xi\right)\) for the associated fiber bundle is well defined (i.e., if \(\llbracket p^{\prime}, \xi^{\prime} \rrbracket=\) \(\llbracket p, \xi \rrbracket\) then \(\left.\phi_{u}\left(\llbracket p^{\prime}, \xi^{\prime} \rrbracket\right)=\phi_{u}(\llbracket p, \xi \rrbracket)\right)\) and bijective.

Short answer

The map \( \phi_u \) is well defined and bijective.

Step-by-step solution

Show that \(\phi_u\) is well defined

To show that \( \phi_u \) is well defined, we need to verify that if two elements, \( \llbracket p, \xi \rrbracket \) and \( \llbracket p', \xi' \rrbracket \), from the total space are in the same equivalence class (that is, \( \llbracket p, \xi \rrbracket = \llbracket p', \xi' \rrbracket \)), then \( \phi_u \) maps them to the same element in \( U \times F \). Given \( \llbracket p, \xi \rrbracket = \llbracket p', \xi' \rrbracket \), there exists an open set \( V \subseteq U \) such that \( p, p' \in V \) and \( g_{uv}(p)\xi = g_{uv}(p')\xi' \) for all local trivializations belonging to \( V \), where \( g_{uv} \) are the transition functions of the fiber bundle. Applying \( \phi_u \) to both sides yields \( \phi_u(\llbracket p, \xi \rrbracket) = \phi_u(\llbracket p', \xi' \rrbracket) \). Thus, \( \phi_u \) is well defined.

Show that \(\phi_u\) is injective

To prove \( \phi_u \) is injective, we need to show that if \( \phi_u(\llbracket p, \xi \rrbracket) = \phi_u(\llbracket p', \xi' \rrbracket) \), then \( \llbracket p, \xi \rrbracket = \llbracket p', \xi' \rrbracket \). We have \( \phi_u(\llbracket p, \xi \rrbracket) = (\pi(p), s_u(p) \xi) = (\pi(p'), s_u(p') \xi') = \phi_u(\llbracket p', \xi' \rrbracket) \). This gives us \( p = p' \) and \( s_u(p) \xi = s_u(p') \xi' \), and applying the inverse of \( s_u(p) \) to both sides, we obtain \( \xi = \xi' \). Hence, \( \llbracket p, \xi \rrbracket = \llbracket p', \xi' \rrbracket \), so \( \phi_u \) is injective.

Show that \(\phi_u\) is surjective

To prove \( \phi_u \) is surjective, we need to show that for any \( (q, f) \in U \times F \), there exists \( \llbracket p, \xi \rrbracket \) in the pre-image of \( U \) of total space such that \( \phi_u(\llbracket p, \xi \rrbracket) = (q, f) \). Let \( q \in U \) and \( f \in F \), if we let \( p = \pi^{-1}(q) \) and take \( \xi = s_u^{-1}(p)f \), then \( \phi_u(\llbracket p, \xi \rrbracket) = (q, s_u(p)s_u^{-1}(p)f) = (q, f) \). Thus, \( \phi_u \) is surjective.

Key concepts

Associated Fiber Bundle

An associated fiber bundle is a significant concept in topology that extends the idea of fiber bundles. In simple terms, it associates a new type of fiber with each point of the base space, connecting them in a structured way. Such constructions arise by associating fibers through an equivalence relation derived from a principle bundle.

Imagine you have a principal bundle, which is like a collection of fibers "stretched" over a base space. Each fiber here is linked to a symmetry (or structure) group. When we create an associated fiber bundle, we effectively replace each fiber with a more complex structure, specified by a representation of this group.

This concept allows us to study complex geometrical and topological structures seamlessly. For instance, the associated fiber bundle helps in scenarios such as describing how vector or tensor fields transform under different coordinate systems.

• Each put-together fiber retains the variability and structure of the principal bundle's fibers, but potentially with a different "shape" or functionality.
• They accommodate additional features like metrics or spin structures on manifolds.
• Associated bundles arise naturally in gauge theories, crucial for advances in physics.

Transition Functions

Transition functions play a key role in the construction of fiber bundles, including associated fiber bundles.

A fiber bundle can be likened to a cover over a space that is not uniform throughout but instead is made up of "patches." Transition functions describe how to "glue" these patches together. They help one understand how the fibers are attached to the base space and how they transform from one coordinate chart to another.

For the associated fiber bundle, transition functions ensure these transformations are carried out consistently across the principal bundle, ensuring the overall bundle structure remains intact.

• They are functions that encode how local trivializations of a bundle overlap and transform over base space intersections.
• In mathematical terms, they are derived from the structure group acting on fibers.
• They must satisfy specific conditions like the cocycle condition for consistent assembly of the bundle.

Local Trivializations

Local trivializations simplify working with fiber bundles by making them look locally simple. Think of this as a tool that "straightens out" parts of a bundle to make it appear like a simple collection of the base space crossed with the fiber.

In a fiber bundle, a local trivialization is like taking a small region of the base and "flattening" it so that the bundle over this region resembles a direct product of the base space with a fiber. This process makes complex spaces manageable and understandable by providing a "local view". Such patches simplify calculations and visualizations of bundles, as they allow one to think locally, not globally.

• They provide a view where the bundle looks like a direct product, offering clarity in complex geometries.
• Serve as a tool for transitioning between local and global perspectives on bundles.
• Are crucial for defining smooth structures and analyzing continuity in bundles.