Regarding fibrations between smooth manifolds

In summary, the two conditions for a quotient map are that U is open iff p^{-1}(U) is open in X and that f is smooth iff f ◦ p is a smooth function on p^{-1}(U).
  • #1
aalma
46
1
Definitions:
1. A map ##p : X → Y## of smooth manifolds is called a trivial fibration with fiber ##Z## which is also a smooth manifold, if there is a diffeomorphism ##θ : X → Y ×Z## such that ##p## is the composition of ##θ## with the natural projection ##pr_1:Y × Z → Y##.
2. A map ##p: X →Y## is a locally trivial fibration with fiber ##Z## if for all ##y \in Y## there exists an open neighborhood ##U⊂Y## of ##y## such that ##p: p^{−1}(U) → U## is a trivial fibration with fiber ##Z##.
3. A smooth map ##p : X → Y## is called a quotient map if the following conditions are fulfilled:
1. ##U ⊂Y## is open iff ##p^{−1}(U)## is open in X.
2. ##f : U →R## is smooth iff ##f ◦p## is a smooth function on ##p^{−1}(U)##.
Now, I need to show that any locally trivial fibration is a quotient map.

Let ##p:X\to Y## a locally trivial fibration.
Well I need to verify the two conditions for a quotient map
  1. ##U ⊂ Y## is open iff ##p^{-1}(U)## is open in ##X##.
  2. ##f : U → R## is smooth iff ##f ◦ p## is a smooth function on ##p^{-1}(U)##.
For the first condition, suppose that ##U## is an open subset of ##Y##. We want to show that ##p^{-1}(U)## is open in ##X##. By the definition of a locally trivial fibration, for every ##y \in U##, there exists an open neighborhood ##V_y## of ##y## such that ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via a diffeomorphism ##θ_y##. Since ##U## is an open subset of ##Y##, we can cover ##U## by the open neighborhoods ##V_y##. That is, ##U = ∪_y V_y##.
Now consider ##p^{-1}(U)##. For any point ##x \in p^{-1}(U)##, we have ##p(x) ∈ U##, so there exists a ##V_y## containing ##p(x)##. Since ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via ##θ_y##, we can write ##x = θ_y(y', z)## for some ##y' \in V_y## and ##z \in Z##. Moreover, since ##V_y## is an open neighborhood of ##y##, we can assume that ##y'## lies in a smaller open set ##U_y## contained in ##V_y##. Therefore, we have ##x = θ_y(y', z) ∈ U_y × Z##, which is contained in ##p^{-1}(V_y)##. Thus, we have shown that every point ##x \in p^{-1}(U)## is contained in an open set of the form ##U_y × Z##, which is diffeomorphic to an open set in ##Y × Z##. Therefore, ##p^{-1}(U)## is an open subset of ##X##.
Does this seem okay?
Next, let's consider the second condition. Suppose that ##f : U → R## is a smooth function on ##U##. We want to show that ##f ◦ p## is a smooth function on ##p^{-1}(U)##. For any point ##x \in p^{-1}(U)##, we have ##p(x) ∈ U##, so there exists a ##V_y## containing ##p(x)##. Since ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via ##θ_y##, we can write ##x = θ_y(y', z)## for some ##y' \in V_y## and ##z \in Z##. Moreover, since ##f## is smooth on ##U##, we can write ##f(y') = g_y(y')## for some smooth function ##g_y## on ##V_y##. Then we have:
##(f ◦ p)(x) = f(p(x)) = f(y') = g_y(y') = (g_y ◦ p')(y', z)##
where ##p'## is the projection ##Y × Z → Y##. We note that ##(g_y ◦ p')## is a smooth function on ##V_y × Z##, which is diffeomorphic to ##p^{-1}(V_y)## via ##θ_y##. Therefore, ##(g_y ◦ p')(y', z)## is a smooth function on ##p^{-1}(V_y)##, which contains ##x##. Since this is true for any ##V_y## containing p(x), we conclude that ##f ◦ p## is a smooth function on ##p^{-1}(U)##.
What do you think?Thanks in advance for any tips..
 
Physics news on Phys.org
  • #2
i apologize if this is unhelpful, but i hope it may be. First step is to realize it suffices to prove this for a trivial fibration p:YxZ-->Y, in which case it follows almost immediately from the definition of the product topology and the fact that p restricts to a diffeomorphism from Yx{q}-->Y, for any point q in Z.
 
  • Like
Likes fresh_42

1. What is a smooth manifold?

A smooth manifold is a mathematical concept used to describe a space that locally resembles Euclidean space. It is a topological space that is locally homeomorphic to Euclidean space, meaning that small portions of the space can be mapped onto a corresponding portion of Euclidean space by a continuous function. Smooth manifolds are important in many areas of mathematics, including differential geometry, topology, and physics.

2. What is a fibration?

A fibration is a mathematical concept that describes a continuous mapping between spaces. Specifically, it is a mapping where each point in the target space has a neighborhood that is mapped homeomorphically onto a neighborhood of the source space. Fibrations are important in many areas of mathematics, including topology, algebraic geometry, and differential geometry.

3. How do fibrations relate to smooth manifolds?

Fibrations between smooth manifolds are mappings that preserve the smooth structure of the manifolds. This means that they are continuous and differentiable, and that they preserve the local Euclidean structure of the manifolds. Fibrations between smooth manifolds are important in understanding the global structure of the manifolds and in studying their topological properties.

4. What is the significance of studying fibrations between smooth manifolds?

Studying fibrations between smooth manifolds can provide insights into the structure and properties of the manifolds themselves. It can also help in understanding the relationships between different manifolds and in classifying them. Additionally, fibrations have applications in many areas of mathematics and physics, making them an important topic of study.

5. What are some examples of fibrations between smooth manifolds?

There are many examples of fibrations between smooth manifolds, including the projection map from a product manifold to one of its factors, the Hopf fibration from the 3-sphere to the 2-sphere, and the tangent bundle projection from a manifold to its base space. Other examples include fibrations over a base space, such as the fibration of a circle over a torus, and fibrations between different types of manifolds, such as a fibration from a complex manifold to a real manifold.

Similar threads

  • Topology and Analysis
2
Replies
43
Views
780
  • Differential Geometry
Replies
20
Views
2K
  • Topology and Analysis
Replies
1
Views
886
  • Topology and Analysis
Replies
9
Views
1K
  • Topology and Analysis
Replies
8
Views
1K
  • Topology and Analysis
Replies
21
Views
2K
  • Topology and Analysis
Replies
17
Views
3K
Replies
5
Views
2K
  • Differential Geometry
Replies
9
Views
408
Replies
2
Views
1K
Back
Top