Tags: #homotopy/bundles #expository #homotopy

Principal Bundles

A principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product \(X × G\) 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, \((x,g) ↦ x\).

A principal \(G\)-bundle, where \(G\) denotes any topological group, is a fiber bundle \(π:P → X\) together with a continuous right action \(P × G → P\) such that \(G\) preserves the fibers of \(P\) (i.e. if \(y ∈ P_x\) then \(yg ∈ P_x\) for all \(g ∈ G\)) and acts freely and transitively on them. This implies that each fiber of the bundle is homeomorphic to the group \(G\) itself.

Since the group action preserves the fibers of \(π:P → X\) 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 \(G{\hbox{-}}\)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.

One of the most important questions regarding any fiber bundle is whether or not it is trivial, i.e. isomorphic to a product bundle. A principal bundle is trivial if and only if it admits a global cross section.

Examples

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

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

\begin{align*}G=\pi _{1}(X)/p_* (\pi_{1}(C))\end{align*}

acts on the fibers of \(p\) via the monodromy action. In particular, the universal cover of \(X\) is a principal bundle over \(X\) with structure group \(\pi_1(X)\) (since the universal cover is simply connected and thus \(\pi_1(C)\) is trivial).

Classification

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 \(EG → BG\).

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 \(B \to BG\).

Classifying Spaces

Definition: A principal \(G{\hbox{-}}\)bundle is a fiber bundle \(F \to E \to B\) in which for each fiber \(\pi^{-1}(b)\coloneqq F_b\), satisfying the condition that \(G\) acts freely and transitively on \(F_b\). In other words, there is a continuous group action \(\curvearrowright: E\times G \to E\) such that for every \(f \in F_b\) and \(g\in G\), we have \(g\curvearrowright f \in F_b\) and \(g\curvearrowright f \neq f\).

Example: A covering space \(\widehat{X} \xrightarrow{p} X\) yields a principal \(\pi_1(X){\hbox{-}}\)bundle.

Remark: A consequence of this is that each \(F_b \cong G \in \text{TopGrp}\) (which may also be taken as the definition). Furthermore, each \(F_b\) is then a homogeneous space, i.e. a space with a transitive group action \(G\curvearrowright F_b\) making \(F_b \cong G/G_x\).

Remark: Although each fiber \(F_b\) is isomorphic to \(G\), there is no preferred identity element in \(F_b\). Locally, one can form a local section by choosing some \(e\in F_b\) to serve as the identity, but the fibers can only be given a global group structure iff the bundle is trivial. This property is expressed by saying \(F_b\) is a \(G{\hbox{-}}\)torsor.

Remark: Every fiber bundle \(F\to E\to B\) is a principal \(\mathop{\mathrm{Aut}}(F){\hbox{-}}\)fiber bundle. Also, in local trivializations, the transition functions are elements of \(G\).

Proposition: A principal bundle is trivial iff it admits a global section. Thus all principal vector bundles are trivial, since the zero section always exists.

Definition: A principal bundle \(F \to E \xrightarrow{\pi} B\) is universal iff \(E\) is weakly contractible, i.e. if \(E\) has the homotopy type of a point.

Definition: Given a topological group \(G\), a classifying space, denoted \(BG\), is the base space of a universal principal \(G{\hbox{-}}\)bundle \begin{align*} G \to EG \xrightarrow{\pi} BG \end{align*} making \(BG\) a quotient of the contractible space \(EG\) by a \(G{\hbox{-}}\)action. We shall refer to this as the classifying bundle.

Classifying spaces satisfy the property that any other principal \(G{\hbox{-}}\)bundle over a space \(X\) is isomorphic to a pullback of the classifying bundle along a map \(X \to BG\).

Let \(I(G, X)\) denote the set of isomorphism classes of principal \(G{\hbox{-}}\)bundles over a base space \(X\), then \begin{align*} I(G, X) \cong [X, BG]_{\text{hoTop}} \end{align*}

So in other words, isomorphism classes of principal \(G{\hbox{-}}\)bundles over a base \(X\) are equivalent to homotopy classes of maps from \(X\) into the classifying space of \(G\).

Proposition: Grassmannians are classifying spaces for vector bundles. That is, there is a bijective correspondence:

\begin{align*} [X, {\operatorname{Gr}}(n, {\mathbf{R}})] \cong \left\{{\text{isomorphism classes of rank $n$ ${\mathbf{R}}{\hbox{-}}$vector bundles over $X$}}\right\} \end{align*}

It is also the case that every such vector bundle is a pullback of the principal bundle \begin{align*} \operatorname{GL}(n, {\mathbf{R}}) \to V_n({\mathbf{R}}^\infty) \to {\operatorname{Gr}}(n, {\mathbf{R}}) \end{align*}