Principal bundle

In mathematics, a principal bundle^[1]^[2]^[3]^[4] is a mathematical object that formalizes some of the essential features of the Cartesian product $X \times G$ of a space $X$ with a group $G$ . In the same way as with the Cartesian product, a principal bundle $P$ is equipped with

An action of $G$ on $P$ , analogous to $(x, g) h = (x, gh)$ for a product space.
A projection onto $X$ . For a product space, this is just the projection onto the first factor, $(x, g) \mapsto x$ .

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of $(x, e)$ . Likewise, there is not generally a projection onto $G$ generalizing the projection onto the second factor, $X \times G \to G$ which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.

A common example of a principal bundle is the frame bundle $F E$ of a vector bundle $E$ , which consists of all ordered bases of the vector space attached to each point. The group $G$ in this case is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no natural way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.

Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories.

Formal definition

A principal $G$ -bundle, where $G$ denotes any topological group, is a fiber bundle $π : P \to X$ together with a continuous right action $P \times G \to P$ such that $G$ preserves the fibers of $P$ (i.e. if $y \in P x$ then $yg \in P x$ for all $g \in G$ ) and acts freely and transitively on them. This implies that each fiber of the bundle is homeomorphic to the group $G$ itself. Frequently, one requires the base space $X$ to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of $π : P \to X$ and acts transitively, it follows that the orbits of the $G$ -action are precisely these fibers and the orbit space $P / G$ is homeomorphic to the base space $X$ . Because the action is free, the fibers have the structure of G-torsors. A $G$ -torsor is a space which is homeomorphic to $G$ but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal $G$ -bundle is as a $G$ -bundle $π : P \to X$ with fiber $G$ where the structure group acts on the fiber by left multiplication. Since right multiplication by $G$ on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by $G$ on $P$ . The fibers of $π$ then become right $G$ -torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal $G$ -bundles in the category of smooth manifolds. Here $π : P \to X$ is required to be a smooth map between smooth manifolds, $G$ is required to be a Lie group, and the corresponding action on $P$ should be smooth.

Examples

The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold $M$ , often denoted $F M$ or $GL(M)$ . Here the fiber over a point $x \in M$ is the set of all frames (i.e. ordered bases) for the tangent space $T x M$ . The general linear group $GL(n, R)$ acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal $GL(n, R)$ -bundle over $M$ .

Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group $O(n)$ . The example also works for bundles other than the tangent bundle; if $E$ is any vector bundle of rank $k$ over $M$ , then the bundle of frames of $E$ is a principal $GL(k, R)$ -bundle, sometimes denoted $F(E)$ .

A normal (regular) covering space $p : C \to X$ is a principal bundle where the structure group

G=\pi _{1}(X)/p_{*}(\pi _{1}(C))

acts on the fibres of $p$ via the monodromy action. In particular, the universal cover of $X$ is a principal bundle over $X$ with structure group $π 1 (X)$ (since the universal cover is simply connected and thus $π 1 (C)$ is trivial).

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$ -bundle over the (left) coset space $G / H$ . Here the action of $H$ on $G$ is just right multiplication. The fibers are the left cosets of $H$ (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to $H$ ).

Consider the projection $π : S 1 \to S 1$ given by $z \mapsto z 2$ . This principal $Z 2$ -bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal $Z 2$ -bundle over $S 1$ .

Projective spaces provide some more interesting examples of principal bundles. Recall that the $n$ -sphere $S n$ is a two-fold covering space of real projective space $RP n$ . The natural action of $O(1)$ on $S n$ gives it the structure of a principal $O(1)$ -bundle over $RP n$ . Likewise, $S 2 n +1$ is a principal $U(1)$ -bundle over complex projective space $CP n$ and $S 4 n +3$ is a principal $Sp(1)$ -bundle over quaternionic projective space $HP n$ . We then have a series of principal bundles for each positive $n$ :

{\mbox{O}}(1)\to S(\mathbb {R} ^{n+1})\to \mathbb {RP} ^{n}

{\mbox{U}}(1)\to S(\mathbb {C} ^{n+1})\to \mathbb {CP} ^{n}

{\mbox{Sp}}(1)\to S(\mathbb {H} ^{n+1})\to \mathbb {HP} ^{n}.

Here $S (V)$ denotes the unit sphere in $V$ (equipped with the Euclidean metric). For all of these examples the $n = 1$ cases give the so-called Hopf bundles.

Basic properties

Trivializations and cross sections

One of the most important questions regarding any fiber bundle is whether or not it is trivial, i.e. isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

Proposition. A principal bundle is trivial if and only if it admits a global cross section.

The same is not true for other fiber bundles. For instance, Vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let $π : P \to X$ be a principal $G$ -bundle. An open set $U$ in $X$ admits a local trivialization if and only if there exists a local section on $U$ . Given a local trivialization

\Phi :\pi ^{-1}(U)\to U\times G

one can define an associated local section

s:U\to \pi ^{-1}(U);s(x)=\Phi ^{-1}(x,e)\,

where $e$ is the identity in $G$ . Conversely, given a section $s$ one defines a trivialization $Φ$ by

\Phi ^{-1}(x,g)=s(x)\cdot g.

The simple transitivity of the $G$ action on the fibers of $P$ guarantees that this map is a bijection, it is also a homeomorphism. The local trivializations defined by local sections are $G$ -equivariant in the following sense. If we write

\Phi :\pi ^{-1}(U)\to U\times G

in the form

\Phi (p)=(\pi (p),\varphi (p)),

then the map

\varphi :P\to G

satisfies

\varphi (p\cdot g)=\varphi (p)g.

Equivariant trivializations therefore preserve the $G$ -torsor structure of the fibers. In terms of the associated local section $s$ the map $φ$ is given by

\varphi (s(x)\cdot g)=g.

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization $({U i}, {Φ i})$ of $P$ , we have local sections $s i$ on each $U i$ . On overlaps these must be related by the action of the structure group $G$ . In fact, the relationship is provided by the transition functions

t_{ij}=U_{i}\cap U_{j}\to G\,

For any $x \in U i \cap U j$ we have

s_{j}(x)=s_{i}(x)\cdot t_{ij}(x).

Characterization of smooth principal bundles

If π : P → X is a smooth principal G-bundle then G acts freely and properly on P so that the orbit space P/G is diffeomorphic to the base space X. It turns out that these properties completely characterize smooth principal bundles. That is, if P is a smooth manifold, G a Lie group and μ : P × G → P a smooth, free, and proper right action then

P/G is a smooth manifold,
the natural projection π : P → P/G is a smooth submersion, and
P is a smooth principal G-bundle over P/G.

Use of the notion

Reduction of the structure group

Given a subgroup H of G one may consider the bundle $P/H$ whose fibers are homeomorphic to the coset space $G/H$ . If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of P which is a principal H-bundle. If H is the identity, then a section of P itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal G-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from G to H). For example:

A 2n-dimensional real manifold admits an almost-complex structure if the frame bundle on the manifold, whose fibers are $GL(2n,\mathbb {R} )$ , can be reduced to the group $\mathrm {GL} (n,\mathbb {C} )\subset \mathrm {GL} (2n,\mathbb {R} )$ .
An n-dimensional real manifold admits a k-plane field if the frame bundle can be reduced to the structure group $\mathrm {GL} (k,\mathbb {R} )\subset \mathrm {GL} (n,\mathbb {R} )$ .
A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group, $\mathrm {SO} (n)\subset \mathrm {GL} (n,\mathbb {R} )$ .
A manifold has spin structure if and only if its frame bundle can be further reduced from $\mathrm {SO} (n)$ to $\mathrm {Spin} (n)$ the Spin group, which maps to $\mathrm {SO} (n)$ as a double cover.

Also note: an n-dimensional manifold admits n vector fields that are linearly independent at each point if and only if its frame bundle admits a global section. In this case, the manifold is called parallelizable.

Associated vector bundles and frames

If P is a principal G-bundle and V is a linear representation of G, then one can construct a vector bundle $E=P\times _{G}V$ with fibre V, as the quotient of the product P×V by the diagonal action of G. This is a special case of the associated bundle construction, and E is called an associated vector bundle to P. If the representation of G on V is faithful, so that G is a subgroup of the general linear group GL(V), then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL(V) to G. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.

Classification of principal bundles

Any topological group $G$ admits a classifying space $BG$ : the quotient by the action of $G$ of some weakly contractible space $EG$ , i.e. a topological space with vanishing homotopy groups. The classifying space has the property that any $G$ principal bundle over a paracompact manifold B is isomorphic to a pullback of the principal bundle $EG \to BG$ .^[5] In fact, more is true, as the set of isomorphism classes of principal $G$ bundles over the base $B$ identifies with the set of homotopy classes of maps $B \to BG$ .

References

↑ Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6. page 35
↑ Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8. page 42
↑ Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. page 37
↑ Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5. page 370
↑ Stasheff, James D. (1971), "H-spaces and classifying spaces: foundations and recent developments", Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society, pp. 247–272 , Theorem 2

Sources

Bleecker, David (1981). Gauge Theory and Variational Principles. Addison-Wesley Publishing. ISBN 0-486-44546-1.
Jost, Jürgen (2005). Riemannian Geometry and Geometric Analysis ((4th ed.) ed.). New York: Springer. ISBN 3-540-25907-4.
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.
Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.

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