Associated bundle

In mathematics, the theory of fiber bundles with a structure group ${\displaystyle G}$ (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from ${\displaystyle F_{1}}$ to ${\displaystyle F_{2}}$, which are both topological spaces with a group action of ${\displaystyle G}$. For a fiber bundle ${\displaystyle F}$ with structure group ${\displaystyle G}$, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems ${\displaystyle U_{\alpha }}$ and ${\displaystyle U_{\beta }}$ are given as a ${\displaystyle G}$-valued function ${\displaystyle g_{\alpha \beta }}$ on ${\displaystyle U_{\alpha }\cap U_{\beta }}$. One may then construct a fiber bundle ${\displaystyle F'}$ as a new fiber bundle having the same transition functions, but possibly a different fiber.

An example

A simple case comes with the Möbius strip, for which ${\displaystyle G}$ is the cyclic group of order 2, ${\displaystyle \mathbb {Z} _{2}}$. We can take as ${\displaystyle F}$ any of: the real number line ${\displaystyle \mathbb {R} }$, the interval ${\displaystyle [-1,\ 1]}$, the real number line less the point 0, or the two-point set ${\displaystyle \{-1,\ 1\}}$. The action of ${\displaystyle G}$ on these (the non-identity element acting as ${\displaystyle x\ \rightarrow \ -x}$ in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles ${\displaystyle [-1,\ 1]\times I}$ and ${\displaystyle [-1,\ 1]\times J}$ together: what we really need is the data to identify ${\displaystyle [-1,\ 1]}$ to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in ${\displaystyle G}$. The associated bundle construction is just the observation that this data does just as well for ${\displaystyle \{-1,\ 1\}}$ as for ${\displaystyle [-1,\ 1]}$.

Construction

In general it is enough to explain the transition from a bundle with fiber ${\displaystyle F}$, on which ${\displaystyle G}$ acts, to the associated principal bundle (namely the bundle where the fiber is ${\displaystyle G}$, considered to act by translation on itself). For then we can go from ${\displaystyle F_{1}}$ to ${\displaystyle F_{2}}$, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.

This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction. ^[1]

Associated bundles in general

Let ${\textstyle \pi :E\to X}$ be a fiber bundle over a topological space ${\displaystyle X}$ with structure group ${\displaystyle G}$ and typical fibre ${\displaystyle F}$. By definition, there is a left action of ${\displaystyle G}$ (as a transformation group) on the fibre ${\displaystyle F}$. Suppose furthermore that this action is effective. ^[2] There is a local trivialization of the bundle ${\displaystyle E}$ consisting of an open cover ${\displaystyle U_{i}}$ of ${\displaystyle X}$, and a collection of fibre maps $${\displaystyle \varphi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times F}$$such that the transition maps are given by elements of ${\displaystyle G}$. More precisely, there are continuous functions ${\displaystyle g_{ij}\colon U_{i}\cap U_{j}\to G}$ such that$${\displaystyle \psi _{ij}(u,f):=\varphi _{i}\circ \varphi _{j}^{-1}(u,f)={\big (}u,g_{ij}(u)f{\big )},\quad {\text{for each }}(u,f)\in (U_{i}\cap U_{j})\times F\,.}$$

Now let ${\displaystyle F'}$ be a specified topological space, equipped with a continuous left action of ${\displaystyle G}$. Then the bundle associated with ${\displaystyle E}$ with fibre ${\displaystyle F'}$ is a bundle ${\displaystyle E'}$ with a local trivialization subordinate to the cover ${\displaystyle U_{i}}$ whose transition functions are given by$${\displaystyle \psi '_{ij}(u,f')={\big (}u,g_{ij}(u)f'{\big )},\quad {\text{for each }}(u,f')\in (U_{i}\cap U_{j})\times F'\,,}$$where the ${\displaystyle G}$-valued functions ${\displaystyle g_{ij}(u)}$ are the same as those obtained from the local trivialization of the original bundle ${\displaystyle E}$. This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of ${\displaystyle G}$-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the ${\displaystyle g_{ij}}$ transform via the same coboundary.) Hence, by the fiber bundle construction theorem, this produces a fibre bundle ${\displaystyle E'}$ with fibre ${\displaystyle F'}$ as claimed.

Principal bundle associated with a fibre bundle

As before, suppose that ${\displaystyle E}$ is a fibre bundle with structure group ${\displaystyle G}$. In the special case when ${\displaystyle G}$ has a free and transitive left action on ${\displaystyle F'}$, so that ${\displaystyle F'}$ is a principal homogeneous space for the left action of ${\displaystyle G}$ on itself, then the associated bundle ${\displaystyle E'}$ is called the principal ${\displaystyle G}$-bundle associated with the fibre bundle ${\displaystyle E}$. If, moreover, the new fibre ${\displaystyle F'}$ is identified with ${\displaystyle G}$ (so that ${\displaystyle F'}$ inherits a right action of ${\displaystyle G}$ as well as a left action), then the right action of ${\displaystyle G}$ on ${\displaystyle F'}$ induces a right action of ${\displaystyle G}$ on ${\displaystyle E'}$. With this choice of identification, ${\displaystyle E'}$ becomes a principal bundle in the usual sense. Note that, although there is no canonical way to specify a right action on a principal homogeneous space for ${\displaystyle G}$, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group ${\displaystyle G}$ (since this comes from the left action of ${\displaystyle G}$), and isomorphic as ${\displaystyle G}$-spaces in the sense that there is a ${\displaystyle G}$-equivariant isomorphism of bundles relating the two.

In this way, a principal ${\displaystyle G}$-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group ${\displaystyle G}$, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.

Fiber bundle associated with a principal bundle

Let ${\displaystyle \pi \colon P\to X}$ be a principal G-bundle and let ${\displaystyle \rho \colon G\to {\text{Homeo}}(F)}$ be a continuous left action of ${\displaystyle G}$ on a space ${\displaystyle F}$ (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective.

Define a right action of ${\displaystyle G}$ on ${\displaystyle P\times F}$ via ^{[3] [4]}

${\displaystyle (p,f)\cdot g=(p\cdot g,\rho (g^{-1})f)\,.}$

We then identify by this action to obtain the space ${\displaystyle E=P\times _{\rho }F=(P\times F)/G}$. Denote the equivalence class of ${\displaystyle (p,f)}$ by ${\displaystyle [p,f]}$. Note that

${\displaystyle [p\cdot g,f]=[p,\rho (g)f]{\mbox{ for all }}g\in G.}$

Define a projection map ${\displaystyle \pi _{\rho }\colon E\to X}$ by ${\displaystyle \pi _{\rho }([p,f])=\pi (p)}$. Note that this is well-defined.

Then ${\displaystyle \pi _{\rho }\colon E\to X}$ is a fiber bundle with fiber ${\displaystyle F}$ and structure group ${\displaystyle G}$. The transition functions are given by ${\displaystyle \rho (t_{ij})}$ where ${\displaystyle t_{ij}}$ are the transition functions of the principal bundle ${\displaystyle P}$.

This construction can also be seen categorically. More precisely, there are two continuous maps ${\displaystyle P\times G\times F\to P\times F}$, given by acting with ${\displaystyle G}$ on the right on ${\displaystyle P}$ and on the left on ${\displaystyle F}$. The associated vector bundle ${\displaystyle P\times _{\rho }F}$ is then the coequalizer of these maps.

Reduction of the structure group

Further information: reduction of the structure group

The companion concept to associated bundles is the reduction of the structure group of a ${\displaystyle G}$-bundle ${\displaystyle B}$. We ask whether there is an ${\displaystyle H}$-bundle ${\displaystyle C}$, such that the associated ${\displaystyle G}$-bundle is ${\displaystyle B}$, up to isomorphism. More concretely, this asks whether the transition data for ${\displaystyle B}$ can consistently be written with values in ${\displaystyle H}$. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

Examples of reduction

Examples for vector bundles include: the introduction of a metric resulting in reduction of the structure group from a general linear group ${\displaystyle \mathrm {GL} (n)}$ to an orthogonal group ${\displaystyle \mathrm {O} (n)}$; and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group ${\displaystyle \mathrm {GL} (2n,\mathbb {R} )}$ to complex general linear group ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$.

Another important case is finding a decomposition of a vector bundle ${\displaystyle V}$ of rank ${\displaystyle n}$ as a Whitney sum (direct sum) of sub-bundles of rank ${\displaystyle k}$ and ${\displaystyle n-k}$, resulting in reduction of the structure group from ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ to ${\displaystyle \mathrm {GL} (k,\mathbb {R} )\times \mathrm {GL} (n-k,\mathbb {R} )}$.

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.

See also

• Spinor bundle

References

1. All of these constructions are due to Ehresmann (1941-3). Attributed by Steenrod (1951) page 36
2. Effectiveness is a common requirement for fibre bundles; see Steenrod (1951). In particular, this condition is necessary to ensure the existence and uniqueness of the principal bundle associated with ${\displaystyle E}$.
3. Husemoller, Dale (1994), p. 45.
4. Sharpe, R. W. (1997), p. 37.

Books

• Steenrod, Norman (1951). The Topology of Fibre Bundles . Princeton: Princeton University Press. ISBN   0-691-00548-6.
• Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN   978-0-387-94087-8.
• Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN   0-387-94732-9.