Trivial fiber bundle vs product space

[cianfa72]

TL;DR

Is the trivial fiber bundle of topological spaces the same as the product space ?

I've a doubt about the following: take the trivial fiber bundle of the base space $B$ and the fiber $F$ vs the product $B \times F$ of topological spaces $B$ and $F$.

Are they really the same ?

As far as I can tell, in the trivial fiber bundle each fiber in the bundle in a distinct/separated copy of the topological space called "the fiber space" $F$, whereas in the product $B \times F$ it is the same copy of $F$ that, let me say, enters multiple times in the product (i.e. all the copies of $F$ are actually identified).

Then, it's true that they are canonically identified, but that's another story....

 

[martinbn]

What is the definition of a trivial bundle that you have seen?

 

[cianfa72]

What is the definition of a trivial bundle that you have seen?

https://en.wikipedia.org/wiki/Fiber_bundle#Trivial_bundle

I'd add that my doubt comes from this picture of $\mathbb E^1 \times \mathbb E^3$ in Penrose book:

 

[martinbn]

https://en.wikipedia.org/wiki/Fiber_bundle#Trivial_bundle

Seems quite clear and unambiguous.

I'd add that my doubt comes from this picture of $\mathbb E^1 \times \mathbb E^3$ in Penrose book:

What is your doubt?

 

[cianfa72]

I believe I got the point. A trivial bundle is different from a global trivializable bundle. The former is a product while the latter is just homeomorphic to a product, however there isn't a canonical/natural homeomorphism.

So $\mathcal A = \mathbb E^1 \times \mathbb E^3$ is a product (i.e. a trivial fiber bundle). It has the two canonical projections on the factors.

Penrose then defines Galilean spacetime $\mathcal G$ as a (trivializable) affine fiber bundle over the base $\mathbb E^1$ and fibers modeled as $\mathbb E^3$. Here there is only the projection on the base space defined by the structure. Since there isn't the other canonical projection of the product, there is no natural way to identify points on the fibers, i.e. absolute time but not absolute space.

See also Aristotelian-vs-galilean-relativity-in-terms-of-bundles

 

[martinbn]

I believe I got the point. A trivial bundle is different from a global trivializable bundle. The former is a product while the latter is just homeomorphic to a product, however there isn't a canonical/natural homeomorphism.

So $\mathcal A = \mathbb E^1 \times \mathbb E^3$ is a product (i.e. a trivial fiber bundle). It has the two canonical projections on the factors.

Penrose then defines Galilean spacetime $\mathcal G$ as a (trivializable) affine fiber bundle over the base $\mathbb E^1$ and fibers modeled as $\mathbb E^3$. Here there is only the projection on the base space defined by the structure. Since there isn't the other canonical projection of the product, there is no natural way to identify points on the fibers, i.e. absolute time but not absolute space.

See also Aristotelian-vs-galilean-relativity-in-terms-of-bundles

I don't think this has anything to do with mathematics. A tribual bundle is a trivial bundle whether it is given as a product or it is diffeomerphic to a product. A spesific realization as a product may be important to physics, but that is a different question.

 

[cianfa72]

I don't think this has anything to do with mathematics. A trivial bundle is a trivial bundle whether it is given as a product or it is diffeomerphic to a product.

Ok so according to you, from a mathematical viewpoint, there is no difference between a trivial vs. (globally) trivializable fiber bundle.

I'd say a product has got a richer structure than a trivial bundle consisting of both canonical/natural projections (to the first and to the second factor). In some sense, once the trivial bundle is defined from the product, then one of the two canonical projections is essentially "forgotten".

 

[martinbn]

Ok so according to you, from a mathematical viewpoint, there is no difference between a trivial vs. (globally) trivializable fiber bundle.

I'd say a product has got a richer structure than a trivial bundle consisting of both canonical/natural projections (to the first and to the second factor). In some sense, once the trivial bundle is defined from the product, then one of the two canonical projections is essentially "forgotten".

I don't understand your point. If it is diffeomerphic to a product you can tranfer any property of the product to it. It has the exact same structure.

 

[cianfa72]

If it is diffeomerphic to a product you can tranfer any property of the product to it. It has the exact same structure.

Yes however, even if diffeomorphic to a product, there is no canonical/natural diffeomorphism. That's the difference with a product that has canonical projections on the factors.

 

[martinbn]

Yes however, even if diffeomorphic to a product, there is no canonical/natural diffeomorphism. That's the difference with a product that has canonical projections on the factors.

What is canonical about a product? A product $A\times B$ can be written as a product in many ways, say as $A\times f(B)$ for any diffeomorphism $f$ of $B$. Just because you start with a chosen one it doesn't make it canonical.

 

[cianfa72]

What is canonical about a product? A product $A\times B$ can be written as a product in many ways, say as $A\times f(B)$ for any diffeomorphism $f$ of $B$. Just because you start with a chosen one it doesn't make it canonical.

Yes. Nevertheless do you agree that a product has got two canonical/natural projections ? A globally trivializable fiber bundle has only one "built in" projection (the projection on base space).

 

[martinbn]

Yes. Nevertheless do you agree that a product has got two canonical/natural projections ? A globally trivializable fiber bundle has only one "built in" projection (the projection on base space).

No, I don't agree because there is nothing canonical about it. If $X=A\times B$ and $X=A\times B'$, which one is the canonical?

 

[cianfa72]

No, I don't agree because there is nothing canonical about it. If $X=A\times B$ and $X=A\times B'$, which one is the canonical?

So I believe what I call trivial fiber bundle is what for instance Lee - Introduction to smooth manifolds calls product fiber bundle $A \times B$.

A trivial (or trivializable) fiber bundle $(E, A,\pi, F)$ is only diffeomorphic to $A \times F$ in such a way that $\pi$ agrees with the projection on the first factor.

 

[martinbn]

So I believe what I call trivial fiber bundle is what for instance Lee - Introduction to smooth manifolds calls product fiber bundle $A \times B$.

A trivial (or trivializable) fiber bundle $(E, A,\pi, F)$ is only diffeomorphic to $A \times F$ in such a way that $\pi$ agrees with the projection on the first factor.

Yes, but what is the difference? You say that one is canonical and the other not, but in what sense. If you have that $A\times B$ is diffeomorphic to $A\times F$, which one is the canonical?

 

[cianfa72]

Yes, but what is the difference? You say that one is canonical and the other not, but in what sense. If you have that $A\times B$ is diffeomorphic to $A\times F$, which one is the canonical?

Maybe I can't explain my point. If I give you explicitly $A \times B$ this naturally includes the projections on the factors. It is true that $A \times B$ is diffeomerphic to $A \times F$ through the diffeomorphism $\phi$, however one is making an arbitrary choice by picking $\phi$.

 

[martinbn]

Maybe I can't explain my point. If I give you explicitly $A \times B$ this naturally includes the projections on the factors. It is true that $A \times B$ is diffeomerphic to $A \times F$ through the diffeomorphism $\phi$, however one is making an arbitrary choice by picking $\phi$.

Yes, but you giving me $A \times B$ involves a choice and is not canonical. It is the same if I gave you explicitely a vector space with a basis, or a manifold and some coordiantes.

 

[cianfa72]

Yes, but you giving me $A \times B$ involves a choice and is not canonical. It is the same if I gave you explicitely a vector space with a basis, or a manifold and some coordiantes.

Sorry, I can't understand. If I give you explicitly $A \times B$, I'm giving you a product that can be treated as a (product) fiber bundle in which the bundle projection is the projection on the first factor.

What is the implied choice you are talking about giving you $A \times B$ ?

 

[martinbn]

Sorry, I can't understand. If I give you explicitly $A \times B$, I'm giving you a product that can be treated as a (product) fiber bundle in which the bundle projection is the projection on the first factor.

What is the implied choice you are talking about giving you $A \times B$ ?

I also don't understand what you mean by "I give you a product" and how it is different from "I give you something that is diffeomeorphic to a product". Say you give me $\mathbb R^1\times \mathbb R^1$, how is that different than giving me $\mathbb R^2$.

 

[cianfa72]

I also don't understand what you mean by "I give you a product" and how it is different from "I give you something that is diffeomeorphic to a product". Say you give me $\mathbb R^1\times \mathbb R^1$, how is that different than giving me $\mathbb R^2$.

By definition $\mathbb R^2$ as set is $\mathbb R^1 \times \mathbb R^1$. Furthermore as topological space it carries the product topology from the standard topology of $\mathbb R^1$. Therefore, to me, $\mathbb R^2$ is exactly $\mathbb R^1 \times \mathbb R^1$ with the relevant topology, i.e. it is not just homeomorphic/diffeomorphic to the latter.

When I say "I'm giving you something that is diffeomorphic to a product" I'm not assuming/giving you any specific diffeomorphism.

 

[martinbn]

By definition $\mathbb R^2$ as set is $\mathbb R^1 \times \mathbb R^1$. Furthermore as topological space it carries the product topology from the standard topology of $\mathbb R^1$. Therefore, to me, $\mathbb R^2$ is exactly $\mathbb R^1 \times \mathbb R^1$ with the relevant topology, i.e. it is not just homeomorphic/diffeomorphic to the latter.

When I say "I'm giving you something that is diffeomorphic to a product" I'm not assuming/giving you any specific diffeomorphism.

You see, here you don't see a difference. So what is the difference then? Let me give you another example. Take a torus say given explicitely by an equation or as $S^1\times S^1$. When you give me a torus which one do you give me, and how are the two different?

 

[cianfa72]

Sorry, I'm not that skilled in this area. What I can say is to take a look at Lee - ISM chapter 10 Ex. 10.37.

As far as I can tell, in a) what he calls product fiber bundle is just an example/instance of trivial (or global trivializable) fiber bundle. In the latter any homeomorphism such that the bundle projection agrees with the projection on the first factor is valid/admissible.