In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber. In the trivial case, is just and the map is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to is called a section of Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber . (Wikipedia).
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will h
From playlist What is a Manifold?
Introduction to Fiber Bundles part 2: Structure Groups
This is an important notion where we the transition functions of a certain fiber bundles lie in a smaller subgroup. This is important for setting up Streenrod's theorem.
From playlist Fiber bundles
The TRUTH about TENSORS, Part 9: Vector Bundles
In this video we define vector bundles in full abstraction, of which tangent bundles are a special case.
From playlist The TRUTH about TENSORS
Introduction to Fiber Bundles Part 3: Associated Bundles and Amalgamated Products
This is an incomplete introduction here. The basic idea is that the associated principal bundle knows all. This should be obvious since all bundles with G-structure are classified by H^1(X,G) --- it turns out you can recover your original bundle from a principal bundle by taking "amalgamat
From playlist Fiber bundles
Algebraic Topology - 12.2 - Fiber Bundles
From playlist Algebraic Topology
algebraic geometry 21 Projective space bundles
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers projective space bundles, with Hirzebruch surfaces and scrolls as examples. It also includes a brief discussion of abstract varieties. Typo: in the definition o
From playlist Algebraic geometry I: Varieties
Introduction to Fiber Bundles Part 4: Torsor Interlude
Torsors and Principal Homogeneous Spaces. What is the difference?
From playlist Fiber bundles
What is a Tensor? Lesson 16: The metric tensor field
What is a Tensor? Lesson 16: The metric tensor field
From playlist What is a Tensor?
B. Berndtsson - The curvature of (higher) direct images
I will first discuss some earlier work on the curvature of direct images of adjoint line bundles under a smooth proper fibration, or more generally a surjective map and (maybe) some of its applications. Then I will present a general formula for the curvature of higher direct images. Th
From playlist Complex analytic and differential geometry - a conference in honor of Jean-Pierre Demailly - 6-9 juin 2017
Benson Farb, Part 2: Surface bundles, mapping class groups, moduli spaces, and cohomology
29th Workshop in Geometric Topology, Oregon State University, June 29, 2012
From playlist Benson Farb: 29th Workshop in Geometric Topology
The TRUTH about TENSORS, Part 10: Frames
What do the octonions have to do with spheres? Skip to the end of the video to find out!
From playlist The TRUTH about TENSORS
Michael Farber (2/24/22): Topological complexity of spherical bundles
I will start by describing the concept of a parametrized motion planning algorithm which allows to achieve high degree of flexibility and universality. The main part of the talk will focus on the problem of understanding the parametrized topological complexity of sphere bundles. I will exp
From playlist Topological Complexity Seminar
Weil-Petersson currents by Georg Schumacher
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
An introduction to spectral data for Higgs bundles.. by Laura Schaposnik
Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio
From playlist Higgs Bundles
Introduction to Fiber Bundles Part 5.2: Steenrod's Theorem (Proof)
This is about reductions of structure groups of fiber bundles. There is a nice way to parametrize the reductions.
From playlist Fiber bundles