Motivation for construction of associated fiber bundle from a principal bundle

Question

Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fiber bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.

I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.

Idea is to construct a fiber bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $\pi_E:E\rightarrow M$ that gives a fiber bundle with fiber $F$.

Kobayashi's proof goes as follows :

They consider the product manifold $P\times F$ with an action of $G$ as $g.(u,\xi)=(ug,g^{-1}\xi)$. Then they consider the quotient space $(P\times F)/G$ and call this $E$.

Consider the projection map $P\times F\rightarrow M$ defined as $(p,\xi)\mapsto \pi(p)$.

This induces a map $\pi_E:E=(P\times F)/G\rightarrow M$. As $P\rightarrow M$ is a principal $G$ bundle given $x\in M$ there exists an open set $U$ containing $x$ and a local trivialization $\pi^{-1}(U)\rightarrow U\times G$. They then give a bijection $\pi_E^{-1}(U)\rightarrow U\times F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they call $(E,\pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.

I am trying to understand the motivation for the above construction.

Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie group $\phi:G\rightarrow H$ with $G\times H\rightarrow H$ given by $(g,h)\mapsto \phi(g)^{-1}h$ do get a principal $H$ bundle in above construction?

Edit: I think above content looks like it is asking why do we need the construction of associated fiber bundles. No, what I am asking is, suppose I have a Principal $G$ bundle $P(M,G)$ with an action of $G$ on a manifold $F$ and I want to associate some fiber bundle on $M$ with fiber $F$. Then, what suggests you to think of above construction? How does it occur naturally? Are there any other properties of Fiber bundles I should have in mind which suggest this way of construction?

Answer 1

This construction reverses the construction of the frame bundle from a vector bundle (e.g., the tangent bundle). The idea is that each point $f \in F_p$ in the frame bundle of a vector bundle $E$ is, by definition a basis of $E_p$. This therefore defines a natural map of $F \times \mathbb{R}^k \rightarrow E$, where $k$ is the rank of $E$. The inverse image of any $e_p \in E$ is naturally isomorphic to $G$, so the quotient $(F\times \mathbb{R}^k)/G$ is a vector bundle isomorphic to $E$.

Answer 2

To answer your specific question, you get a principal $H$-bundle, often denoted $P \times^G H$. The transition maps are clearly just given by applying $\phi$ to those of $P$.

I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.

Answer 3

Maybe it also makes sense to provide an "abstract" answer to this. Place yourself in the category of spaces over $M$. For present purposes it does not need to be a manifold, and in fact the category we work in may be more general too. I will suppress $M$, i. e. when I say "an object $X$" I will actually mean a map $X\to M$, and when I say "a morphism $Y\to Z$" I will actually mean a fibrewise map from $Y\to M$ to $Z\to M$, etc.

Now for an object $X$, there is another one $\operatorname{Aut}(X)$, with the property that morphisms from any $Y$ to $\operatorname{Aut}(X)$ are in one-to-one correspondence with fibrewise automorphisms of $Y\times_MX\to Y$ over $Y$. This has canonical group structures on its fibres; it might be a group bundle (but not necessarily) and this group bundle might even be trivial, i. e. be isomorphic (over $M$) to the projection $M\times G\to M$ for some "usual" group $G$, and then one gets "your" situation.

More generally, for another object $X'$, there is still another one $\operatorname{Iso}(X,X')$ such that fibrewise maps from any $Y$ to $\operatorname{Iso}(X,X')$ are in one-to-one correspondence with fibrewise isomorphisms between $Y\times_MX\to Y$ and $Y\times_MX'\to Y$ over $Y$.

Now $\operatorname{Iso}(X,X')$ comes with a canonical right $\operatorname{Aut}(X)$-action and a canonical left $\operatorname{Aut}(X')$-action which commute, and would be a principal left $\operatorname{Aut}(X')$- and a principal right $\operatorname{Aut}(X)$-bundle in appropriate sense, except that it may happen not to have global support - i. e. the map $\operatorname{Iso}(X,X')\to M$ may happen not to be surjective (in fact it might well happen that $\operatorname{Iso}(X,X')$ is empty).

However if that map is surjective, then you may try to reconstruct $X'$ from $X$ together with $\operatorname{Iso}(X,X')$. More precisely it is natural to hope that in this case $X'$ is isomorphic to the (fibrewise) quotient of $\operatorname{Iso}(X,X')\times X$ by $\operatorname{Aut}(X)$; informally speaking, you take pairs $\left\langle\varphi,x\right\rangle$ where $x\in X$ and $\varphi:X\to X'$, and identify $\left\langle\varphi,\alpha(x)\right\rangle$ with $\left\langle\varphi\circ\alpha,x\right\rangle$ for $\alpha\in\operatorname{Aut}(X)$. This is the associated bundle construction.

All this works for sure when $X$ is a trivial bundle, i. e. is the projection $M\times F\to M$ for some $F$; then (in good cases) $\operatorname{Aut}(X)$ will be the trivial group bundle $M\times G\to M$. Given some structure on $F$ (such that a vector space) you might want to further switch to the subgroup consisting of structure preserving automorphisms, and to the subbundle of $\operatorname{Iso}(X,X')$ consisting of structure preserving (fibrewise) isomorphisms. You then get as one of the examples what you are after: a vector bundle is more or less the same thing as a fibrewise vector space $E\to M$ such that for a trivial one $M\times V\to M$ the object $\operatorname{Iso}(M\times V,E)$ has global support; the latter can then be identified with the frame bundle of $E$ once you choose a basis in $V$.

I cannot think of an appropriate reference right away but there surely are many, all this is well known at least from 1970ies on...

Answer 4

This is just an attempt to elaborate a bit on Ben McKay's answer beyond the confines of a mere comment.

Principal $G$-bundles $P(M,G)$ over $M$ can be understood as a sort of "universal generator" of transition cocycles for its associated $G$-bundles over $M$. More precisely, the transition cocycles associated to a $G$-bundle atlas for $P(M,G)$ are transition cocycles for the associated $G$-bundle with any given typical fiber $F$, up to the choice of $G$-action on $F$. The quotient $(P\times F)/G$ is meant to achieve precisely that. This is especially clear in the case of frame bundles and vector bundles, as depicted in Deane Yang's answer.

Since the family of transition cocycles encodes all information on the topological deviations of $P\times_G F$ from the trivial bundle $M\times F$, this means that all nontrivial topological information of $P\times_G F$ in this sense is already encoded in $P(M,G)$.

Answer 5

I'm late, but I think the answer can be given in a simpler fashion.

A principal bundle is in particular a fibre bundle, with the extra restriction that the fibre has to be a group $G$. To me, the most direct way of again relaxing to an arbitrary fibre $F$ in $P(M,G)$ is to first enlarge $P$ to be a trivial fibre bundle (with fibres $F$) like $P\times F$ and then dividing out the unwanted groupy fibres $G$: $$E:=(P\times F)/G\quad\text{meaning we identify}\quad (u,f)\sim(ug,g^{-1}.f).$$ This is like if you want $5$ to be $7$, you boldly write $$7=(5\times 7)/5.\qquad\text{(sorry)}$$

Answer 6

Here is a physical interpretation of the associated bundle.

In gauge field theory, the symmetry group $G$ of a principal bundle $G\rightarrow P\rightarrow M$ is interpreted as the thing that dictates the possible interactions of elementary particles, in the following way. Given a representation of $G$ on a vector space $V$, one may construct the associated vector bundle as the orbit space $E=(P\times V)/G$ with respect to the action you defined in your question. Now $E$ comes equipped with a natural structure of a vector bundle. The sections of this vector bundle are interpreted as the states of the particles that your model describes (e.g. the scalar Higgs boson). When describing the force carriers, you utilize the notion of a connection on your principal bundle. These are 1-forms on $P$; pulled back (via a local section of $P$), they are interpreted as the states of the force-carriers that your model describes (e.g. the weak W- and Z-bosons).

This is the mathematical setting for describing the physical interactions between force carriers and scalar bosons (both are bosons, the former are the so-called gauge bosons): equipping your principal bundle with a connection, you end up with a natural construction of a covariant derivative on $E$, which describes the coupling of elementary particles. You even end up with some relevant Feynman diagrams. Here wiggly lines correspond to gauge bosons, and dashed lines to scalar bosons.

In this way, we may describe how force carriers obtain mass, via the so-called mechanism of "spontaneous symmetry breaking". Finally, interactions of bosons and fermions are described by introducing additionaly a spin structure on your base space $M$.