What exactly is a trivial bundle?

Question

I (think I) understand what the definition of the fibre bundle $(E,\pi,M,F,G)$ is, where $E$ is the total space, $\pi$ is the projection, $M$ is the base space, $F$ is the typical fibre, and $G$ is the structure. I've read many sources that state a bundle is a trivial bundle if it homeomorphic/diffeomorphic to the product space $M \times F$.

Now, my question is perhaps really silly, but what exactly is homeomorphic/diffeomorphic to $M \times F$? In other words, letting $\sim$ denote the homeomorphic/diffeomorphic relation, then we should be able to write:

\(M \times F \sim \; ?\)

and what does the $?$ signify? Should it be $(E,\pi,M,F,G)$, or $E$, or something else?

As a bonus, but not the main question: why are physicists interested in the trivial bundle?

Answer 1

"Homeomorphic / diffeomorphic" means that there is a bundle map $\phi:E\rightarrow M\times F$ from $\pi$ to $pr_1$ covering the identity map $id_M:M\ni x\mapsto id_M(x)=x\in M$ on the base manifold - that is, $\phi$ is such that $pr_1\circ\phi=id_M\circ\pi=\pi$ (sorry, I don't know how to do commutative diagrams here in Physics Overflow) - and such that $\phi$ is a homeomorphism / diffeomorphism of the total spaces. Here $pr_1(x,q)=x$ ($x\in M,q\in F$) can be seen as the projection map of the bundle $(M\times F,pr_1,M,F,G)$; recall that $G$ acts on the fiber $F$, and this action can be lifted to $M\times F$ - namely, $g\cdot(x,q)=(x,g\cdot q)$. We also demand that $\phi$ is *$G$-equivariant*, i.e. it commutes with the respective $G$-actions on each fiber. Such a bundle isomorphism is therefore called a *(global) trivialization*.

The interest in trivial bundles lies in two facts: (1) any fiber bundle (regardless of the presence of a $G$-structure) over a contractible topological space (e.g. $\mathbb{R}^n$ or an interval of the real line) is trivial, but (2) even if that's the case, there is no canonical choice of global trivialization for a trivial bundle. For instance, supposing that $\pi$ is an associated bundle to a principal fiber bundle $(P,\eta,M,G)$ over $M$ (which can always be arranged to be the case), one can compose $\phi$ with a gauge transformation of $\pi$ (i.e. a section of $P\times_G G$ acting on $E$) and get another global trivialization. Let's take a concrete example: a necessary and sufficient condition for the tangent bundle of a (say, smooth) $n$-dimensional manifold $M$ to be trivial (we then say that $M$ is *parallelizable*) is that there are $n$ (smooth) vector fields on $M$ which are linearly independent at each point. However, one can choose many other such $n$-tuples of vector fields by performing gauge transformations as above. In the case $M=\mathbb{R}^n$, one may of course use the global coordinate vector fields as a global trivialization, but sometimes it is convenient to choose others, which mostly cannot be realized as coordinate vector fields coming from another (global) coordinate chart. For instance, choose a gauge transformation which doesn't come from the action of a diffeomorphism (that is, not of the form $X_p\mapsto (Tf(p))^{-1} X_{f(p)}$, where $f:M\rightarrow M$ is a diffeomorphism). For topologically non-trivial, parallelizable manifolds (e.g. a torus), one doesn't even have global coordinate charts to provide a global trivialization of its tangent bundle.

Comments

Thank for your answer and +1 because your answer gives a lot of food for thought. I do have some difficulties (which are completely due to my ignorance ;)). I'm trying to convert your answer to the notation and definitions that my book is using. For instance, my book defines local trivializations as:

\(\phi_i : U_i \times F \to \pi^{-1}(U_i)\)

where $U_i$ is a coordinate neighbourhood and \(\pi : E \to M\). It seems that your definition of $\phi$ is like the inverse of a global trivialization, right? I was also wondering if you could define \(pr_1\) (because my book does not mention this at all)?

I'm also confused because you say $\phi:E \to M \times F$ sends $\pi$ to $pr_1$. Does that mean we can identify $\pi$ with $E$ and $M \times F$ with $pr_1$?

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@Hunter According to your convention, yes (then you can call $\phi^{-1}$ a global trivialization if you wish). I'm using the convention employed in the book of I. Kolár, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry"(Springer-Verlag, 1993), where $\pi^{-1}(U_i)$ is taken as the domain of a local trivialization.

$pr_1$ is actually defined in the first paragraph (line 5) of my answer.

Indeed, you can essentially identify all of the bundle structure with the projection map $\pi$, since the total space $E$ and the base $M$ are respectively the domain and codomain of $\pi$ - the fiber $F$, on its turn, is given up to diffeomorphism / homeomorphism by the inverse image $\pi^{-1}(p)$ for any $p\in M$. That is enough to reconstruct the bundle structure, since all the defining properties of a fiber bundle can be seen as properties of $\pi$. The only thing that cannot be recovered from $\pi$ alone is the $G$-structure, since a fiber bundle may admit many different $G$-structures - coming, for instance, from different $G$-actions on the fiber $F$.

Answer 2

[ Pedro was quicker, but I'll post a reply nevertheless. Maybe it's complementary, being more of a chatty exposition to the point of fiber bundles. ]

Basic Idea of the Definition of Fiber Bundles

One way to think of fiber bundles is that they are the data to globally twist functions (on spacetime, say) where “global twist” is much in the sense of “global anomaly” and the like, namely an effect visible on topologically nontrivial spaces when moving around non-contractible cycles. The concept of monodromy – which may be more familiar to physicists – is closely related: monodromy is something exhibited by aconnection on a bundle and specifically by a flat bundle. For a discrete structure group (gauge group) everybundle is flat, and in this case non-trivial bundles and non-trivial monodromy come down to essentially the same thing (see also at local system).

More explicitly, suppose X denotes spacetime and F denotes some space that one wants to map into. For instance F might be the complex numbers and a free scalar field would be a function X→F. For the following it is useful to talk about functions a bit more indirectly: observe that the projection F×X→X from theproduct of F with X down to X is such that a section of this map is precisely a function X→F. We think of X×F→X as encoding the fact that there is one copy of F associated with each point of X, and think of a function with values in F as something that, of course, takes values in F over each point of X. One says that X×F→X is the trivial F-fiber bundle over X.

The point being that more generally we may add a global “twist” to the F-valued functions by making the spaceF vary a bit as we move along X. For a fiber bundle one requires that it doesn’t change much: in fact the word “fiber” in “fiber bundle” refers to the fact that all fibers (over all points of X) are equivalent. But the point is that any F may be equivalent to itself in more than one way (it may have “automorphisms”), and this allows non-trivial global structure even though all fibers look alike.

In this sense, a general F-fiber bundle on some X is defined to be a space P equipped with a map P→X to the base space X (e.g. to spacetime), such that locally it looks like the trivial F-fiber bundle, up to equivalence. To say this more technically: P→X is called an F-fiber bundle if there exists a cover (open cover) of X by patches (e.g. coordinate charts!) \(U_i \hookrightarrow X\) for some index set I, such that for each patch \(U_i\) (with i∈I) there exists a fiberwise equivalence between the restriction \(P|_{U_i}\) of P to \(U_i\), and the trivial F-fiber bundle \(F \times U_i \to U_i\) over the patch \(U_i\).

To say this again in terms of sections: this means that a section of P is locally on each (coordinate) patch Uisimply an F-valued function,but when we change patches (change coordinates) then there may be a non-trivialgauge transformation that relates the values of the function on one patch to that on another patch, where they overlap.

Even if this may seem a bit roundabout on first sight, this is actually something at the very heart of modern physics, in that it embodies the two central principles of modern physics, namely

1. the principle of locality;

2. the gauge principle.

The first roughly says that every global phenomenon in physics must come from local data. In the above discussion this means that any “globally F-valued thing on spacetime X” must come from just F-valued functions on local (coordinate) charts \(U_i \hookrightarrow X\) of spacetime. BUT – and this is key now –, second, the gauge principle says that we may never strictly identify any two phenomena in physics (neither locally nor globally) but we must always ask instead for gauge transformations connecting two maybe seemingly different phenomena. Hence combining the gauge principle with the locality principle means that if an F-valued something on spacetime is locally given by plain F-valued functions, then it should globally given by gluing these F-valued functions together not by identification but by gauge equivalence. The result may be a structure that has global twists, and the nature of these global twists is precisely what an F-fiber bundle embodies.

Comments

Thanks for your answer! Especially, the part you write about on trivial fibre bundle is really useful for now and I will definetely come back to this answer after having studied it a bit more. I do still find it all quite abstract (unfortunately, I have a more physics than maths background), but I know it is very important because I came across it a lot when I was writing a project on magnetic monopoles last year. Either way, thanks again.