Classifying Spaces

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

In previous posts we introduced characteristic classes to classify vector bundles. One of the most interesting aspects is the manner in which their existence is established: we saw that there is a certain universal bundle defined over a large space, such that any bundle can be realized as its pullback. For instance, for real vector bundles the tautological \(k\)-plane bundle over the infinite real Grassmannian

\[E(\gamma^k_\infty)\rightarrow \Gr(k, \mathbb{R}^\infty)\]

played this role (§Stiefel-Whitney Characteristic Classes, §§Grassmannians), and a similar construction existed for complex vector bundles as well (§Characteristic Classes of Vector Bundles, ⁋Example 8). On the other hand, since the pullback of a vector bundle depends only on the homotopy class of the map, the isomorphism class of a rank \(k\) vector bundle over a fixed space \(B\) is given by the set of homotopy classes of maps into these spaces, \([B, \Gr(k, \mathbb{R}^\infty)]\) (or \([B, \Gr(k, \mathbb{C}^\infty)]\)).

Another perspective on vector bundles was to regard them as collections of transition functions. For example, any vector bundle was determined by specifying

\[g_{ij}: U_i\cap U_j\rightarrow \GL(k;\mathbb{R})\]

on the overlaps of a trivializing open cover, and this perspective was powerful in that, for instance, replacing \(\GL(k;\mathbb{R})\) with \(\GL^+(k;\mathbb{R})\) also accounted for oriented vector bundles.

The goal of this post is to connect these two perspectives. That is, we will more generally define the notion of a principal \(G\)-bundle that arises when the structure group is a (topological) group \(G\), and we will define the classifying space \(\B G\) that classifies these. Throughout this post, \(G\) always denotes a topological group, and unless otherwise stated, the base space is assumed to be paracompact Hausdorff.

Definition of Principal Bundle

In a vector bundle the fibers carry a vector space structure, and its transition functions are linear automorphisms preserving this structure, that is, elements of \(\GL(k;\mathbb{R})\). When dealing with a general structure group \(G\), it is natural to take the fiber itself to be \(G\) and to specify the transition by left translation in \(G\). The slight difference is that from this perspective there is no distinguished point: an element of the fiber that is the identity in one chart is understood, via left translation, as a different element of the fiber in another chart. In other words, these elements should be thought of as \(G\)-torsors, and to describe this in a coordinate-independent manner we use a \(G\)-action on the total space.

Definition 1 For a topological group \(G\), suppose a fiber bundle \(p:P\rightarrow X\) and a continuous right action \(P\times G\rightarrow P\) are given. This data is called a principal \(G\)-bundle if the following three conditions hold.

1. The \(G\)-action preserves fibers. That is, \(p(y\cdot g)=p(y)\) for all \(y\in P\) and \(g\in G\).
2. The \(G\)-action is free and transitive on each fiber. That is, for any \(x\in X\), fixing a point \(y\in p^{-1}(x)\), the map \(g\mapsto y\cdot g\) is a bijection from \(G\) onto \(p^{-1}(x)\).
3. (\(G\)-equivariant local triviality) For each \(x\in X\) there exists an open neighborhood \(U\) and a \(G\)-equivariant homeomorphism \(\varphi:p^{-1}(U)\rightarrow U\times G\) compatible with \(p\) over \(U\). Here the \(G\)-action on \(U\times G\) is given by \((u,h)\cdot g=(u,hg)\).

Thus, locally we attach copies of \(G\) in the fiber direction over the base space, and on each fiber \(G\) acts by right translation.

A morphism between two principal \(G\)-bundles \(P,P'\rightarrow X\) is a continuous map \(f:P\rightarrow P'\) satisfying \(p'\circ f=p\) and compatible with the \(G\)-action; if it is a homeomorphism we call it an isomorphism between the two principal bundles. Fixing local trivializations \(\varphi_i:p^{-1}(U_i)\rightarrow U_i\times G\), over each \(U_i\cap U_j\) there is determined a continuous map \(g_{ij}:U_i\cap U_j\rightarrow G\) satisfying \(\varphi_i\circ\varphi_j^{-1}(u,h)=(u,g_{ij}(u)h)\). These transition functions satisfy the same cocycle condition as in vector bundles,

\[g_{ij}(x)g_{jk}(x)=g_{ik}(x),\qquad g_{ii}(x)=e,\]

and two cocycles \((g_{ij})\) and \((g_{ij}')\) give the same bundle if and only if there exist continuous maps \(\lambda_i:U_i\rightarrow G\) such that \(g_{ij}'=\lambda_i g_{ij}\lambda_j^{-1}\). Therefore, isomorphism classes of principal \(G\)-bundles trivializing over an open cover \(\mathcal{U}=\{U_i\}\) are classified by (nonabelian) Čech cohomology \(\check{H}^1(\mathcal{U};G)\), and when \(G=\GL(k;\mathbb{R})\) this coincides exactly with the classification of vector bundles from the previous post.

While a vector bundle always has a zero section, in a principal bundle the fiber is not \(G\) but a \(G\)-torsor, so it is not obvious how to choose a section playing the analogous role. Indeed, the following proposition shows that the existence of a section completely determines the triviality of a principal bundle.

Proposition 2 A principal \(G\)-bundle \(p:P\rightarrow X\) is isomorphic to the trivial bundle if and only if there exists a continuous global section \(s:X\rightarrow P\).

Proof

The trivial bundle \(X\times G\) has the section \(x\mapsto(x,e)\), so if \(P\) is trivial we obtain a section via an isomorphism. Conversely, suppose a section \(s:X\rightarrow P\) exists, and consider the map

\[\Phi:X\times G\rightarrow P,\qquad (x,g)\mapsto s(x)\cdot g.\]

Since the \(G\)-action is simply transitive on each fiber (condition (2) of Definition 1), the restriction of \(\Phi\) to each fiber \(\{x\}\times G\rightarrow p^{-1}(x)\) is bijective, and hence \(\Phi\) is bijective. By definition \(\Phi\) satisfies \(p\circ\Phi=\pr_1\) and is compatible with the \(G\)-action, so it is a morphism over \(X\). Finally, that \(\Phi\) is a homeomorphism can be checked locally: in a trivialization \(\varphi:p^{-1}(U)\rightarrow U\times G\), if \(s\) is written as \(x\mapsto(x,\sigma(x))\) (\(\sigma:U\rightarrow G\) continuous), then \(\Phi\) becomes \((x,g)\mapsto(x,\sigma(x)g)\), and its inverse \((x,h)\mapsto(x,\sigma(x)^{-1}h)\) is continuous.

This proposition reveals the decisive difference between principal bundles and vector bundles. Therefore, to relate principal bundles and vector bundles we need an object that mediates between the two.

The simplest reason a principal \(G\)-bundle cannot be a vector bundle is that the fiber of a vector bundle is a vector space, not a group. We resolve this as follows.

Definition 3 Suppose a principal \(G\)-bundle \(p:P\rightarrow X\) and a topological space \(F\) on which \(G\) acts continuously from the left are given. Define a \(G\)-action on \(P\times F\) by

\[(y,f)\cdot g=(y\cdot g,\ g^{-1}\cdot f)\]

and write its orbit space as \(P\times_G F=(P\times F)/G\). Then the map \(P\times_G F\rightarrow X\) induced by \((y,f)\mapsto p(y)\) is a fiber bundle with fiber \(F\), and we call this the associated bundle of \(P\).

Intuitively, this attaches the fiber \(F\) along the twisted structure of the principal \(G\)-bundle; for instance, applying this to the trivial \(G\)-bundle \(X\times G\) with fiber \(F\) yields the trivial fiber bundle \(X\times F\), and similarly applying it to a slightly twisted (that is, non-trivial) principal \(G\)-bundle \(P\) with fiber \(F\) produces a fiber bundle whose fiber is \(F\) and which inherits the twisting data from \(P\).

The most transparent example is a vector bundle, so let us follow the definition step by step here. The topological group \(G=\GL(k;\mathbb{R})\) acts on the left on the dimension \(k\) real vector space \(F=V\). Also, for convenience suppose the trivial \(G\)-bundle \(P=X\times G\) is given. Then the \(G\)-action defined on the product space \(P\times F=(X\times G)\times V\) is

\[\bigl((x,g),v\bigr)\cdot h=\bigl((x, gh),h^{-1}v\bigr)\]

and its orbit space is nothing but \(X\times V\). This is because taking the action by \(h=g^{-1}\) on any element \(((x,g),v)\) gives

\[((x,g),v)\cdot g^{-1}=\bigl((x,e),gv\bigr),\]

and in this process we identified \((x,e)\) in \(X\times G\) with the element \(x\) of \(X\), using that the map

\[[((x,g),v)]\mapsto(x,gv)\]

is a well-defined homeomorphism. That is, starting from a trivial \(G\)-bundle, the associated bundle is also trivial.

More generally, if \(P\) is given by transition functions \(g_{ij}\) over \(\{U_i\}\), then \(P\times_G F\) becomes the bundle with fiber \(F\) over the same \(U_i\) whose transition is given by the action of \(g_{ij}\) on \(F\).

The reverse construction is also possible. Given a rank \(n\) vector bundle \(E\rightarrow X\), the space of all ordered bases, or frames, of the fiber \(E_x\) over each \(x\),

\[\Fr(E)=\{(x,b)\mid x\in X,b\text{ an ordered basis of $E_x$}\},\]

becomes a principal \(\GL(n;\mathbb{R})\)-bundle under the action of \(\GL(n;\mathbb{R})\) that changes the basis by multiplying a matrix on the right, and we call this the frame bundle of \(E\). The following proposition shows that these two constructions are inverse to each other.

Proposition 4 Over a topological space \(X\), there is a natural one-to-one correspondence between isomorphism classes of principal \(\GL(n;\mathbb{R})\)-bundles and isomorphism classes of rank \(n\) real vector bundles. This correspondence assigns to a principal bundle \(P\) the associated bundle \(P\times_{\GL(n;\mathbb{R})}\mathbb{R}^n\), and to a vector bundle \(E\) the frame bundle \(\Fr(E)\).

Proof

It suffices to verify that the two constructions are mutually inverse.

First, by definition a point of the frame bundle \(\Fr(E)\) is the same as an ordered basis of the fiber \(E_x\), and this carries exactly the same information as a linear isomorphism \(b:\mathbb{R}^n\rightarrow E_x\), since it is enough to see where each standard basis vector of the standard Euclidean space \(\mathbb{R}^n\) goes. The morphism defined in this way,

\[\Fr(E)\times_{\GL(n;\mathbb{R})}\mathbb{R}^n\rightarrow E,\qquad [(b,v)]\mapsto b(v),\]

is well defined because changing \((b,v)\) to \((b\circ A, A^{-1}v)\) still gives the same value, \((b\circ A)(A^{-1}v)=b(v)\). Since this morphism is a linear isomorphism on each fiber, it is a vector bundle isomorphism.

Conversely, starting from a principal bundle \(P\), one can verify that the frame bundle of \(P\times_G\mathbb{R}^n\) is again isomorphic to \(P\) by checking that the transition functions \(g_{ij}\) agree on both sides in a local trivialization.

Both constructions preserve transition functions, hence isomorphism classes, and the naturality of the morphism follows from compatibility with pullback.

Thanks to this equivalence, every classification problem for vector bundles is translated into a problem for principal \(\GL(n;\mathbb{R})\)-bundles. In the same way, complex vector bundles correspond to principal \(\GL(n;\mathbb{C})\)-bundles, and oriented real vector bundles correspond to principal \(\GL^+(n;\mathbb{R})\)-bundles. Therefore, if we can classify principal \(G\)-bundles for an arbitrary structure group \(G\), all these cases are solved at once.

Just as with vector bundles, given a continuous map \(f:X'\rightarrow X\) and a principal \(G\)-bundle \(p:P\rightarrow X\), the pullback bundle

\[f^\ast P=\{(x',y)\in X'\times P\mid f(x')=p(y)\}\]

is defined. Giving the action by \((x',y)\cdot g=(x',y\cdot g)\) makes \(f^\ast P\rightarrow X'\) again a principal \(G\)-bundle, and from the perspective of transition functions this corresponds to pulling back \(g_{ij}\) to \(g_{ij}\circ f\). The crucial fact is that this pullback depends only on the homotopy class of \(f\).

Theorem 5 (Homotopy Invariance of Pullback) Suppose \(X\) is paracompact Hausdorff and \(f_0,f_1:X\rightarrow Y\) are homotopic (§Homotopy, ⁋Definition 2). Then for any principal \(G\)-bundle \(p:P\rightarrow Y\), the pullbacks \(f_0^\ast P\) and \(f_1^\ast P\) are isomorphic over \(X\).

Proof

The key is the following fact.

When \(X\) is paracompact Hausdorff, a principal \(G\)-bundle \(Q\) over \(X\times[0,1]\) is isomorphic to the pullback of its restriction to \(X\times\{0\}\) by the projection \(X\times[0,1]\rightarrow X\times\{0\}\).

This is the covering homotopy property of bundles, which follows from the fact that a trivializing cover over a paracompact Hausdorff base admits a locally finite partition of unity. The gist of the proof is to divide \([0,1]\) into small intervals, glue trivializations over each interval, and patch these local isomorphisms together using a partition of unity.

Now let a homotopy \(H:X\times[0,1]\rightarrow Y\) connecting \(f_0,f_1\) be given and define \(Q=H^\ast P\). By the fact above, \(Q\) is isomorphic to the pullback of \(Q\vert_{X\times\{0\}}=f_0^\ast P\) by the projection, and repeating the same argument at the end \(X\times\{1\}\) shows that \(Q\vert_{X\times\{1\}}=f_1^\ast P\) is also isomorphic to the same bundle.

In particular, if \(X\) is contractible then the identity map is homotopic to a constant map, so every principal \(G\)-bundle over \(X\) is trivial. In general, a CW complex is always paracompact Hausdorff, so for the bases we intend to work with, the hypothesis of the above theorem is automatically satisfied.

Universal Bundle and Classifying Space

Theorem 5 (Homotopy Invariance of Pullback) tells us that assigning a function \(f\) to \(f^\ast P\) depends only on the homotopy class of \(f\). Therefore, if we can take some fixed principal \(G\)-bundle as a source from which all other bundles can be obtained by pullback, the classification of principal \(G\)-bundles will reduce to counting homotopy classes into that source space, generalizing the situation in vector bundles where the universal bundle over \(\Gr(k,\mathbb{R}^\infty)\) was such a source.

Definition 6 For a topological group \(G\), a principal \(G\)-bundle \(p:\E G\rightarrow \B G\) is called a universal bundle if the total space \(\E G\) is contractible, that is, \(\E G\) is homotopy equivalent to a point (§Homotopy, ⁋Definition 4). In this case we call the base space \(\B G\) the classifying space of \(G\).

Thus, a universal \(G\)-bundle is a free \(G\)-action on a contractible space, whose orbit space \(\B G=\E G/G\) is the base space and whose projection map is the bundle map. The condition that \(\E G\) is contractible will be used crucially in Theorem 8 (Classification Theorem). Before that, the following holds.

Theorem 7 (Milnor) For any topological group \(G\), a universal bundle \(\E G\rightarrow \B G\) exists.

The proof uses the infinite join of \(G\),

\[\E G=G\ast G\ast G\ast\cdots,\]

and the point is that this space is \(n\)-connected for every \(n\), hence weakly contractible, and under a CW structure it is contractible.

On the other hand, a universal bundle is essentially unique. Suppose two universal bundles \(\E G\rightarrow \B G\) and \(\E G'\rightarrow \B G'\) are given. Since \(\E G'\rightarrow\B G'\) is universal, by Theorem 8 (Classification Theorem) to be shown below there exists a morphism \(u:\B G\rightarrow \B G'\) classifying the principal \(G\)-bundle \(\E G\) over \(\B G\) such that \(\E G\cong u^\ast\E G'\), and swapping the roles of the two bundles likewise gives \(v:\B G'\rightarrow \B G\) with \(\E G'\cong v^\ast\E G\). Then \((v\circ u)^\ast\E G\cong u^\ast\E G'\cong\E G\), but the identity map also classifies \(\E G\), so by the injectivity part of the same theorem \(v\circ u\) is homotopic to the identity map of \(\B G\), and for the same reason \(u\circ v\) is homotopic to the identity map of \(\B G'\). Therefore \(\B G\) is determined without ambiguity beyond homotopy equivalence, and we speak of \(\B G\) as the classifying space.

Even apart from the ingredients of this argument, the most central result of this post is of course the following theorem.

Theorem 8 (Classification Theorem) For a paracompact Hausdorff space \(X\) and a topological group \(G\), let \([X,\B G]\) denote the set of free homotopy classes from \(X\) to \(\B G\). Then the map pulling back the universal bundle \(\E G\rightarrow \B G\),

\[[X,\B G]\rightarrow\{\text{principal $G$-bundles over $X$}\}/{\cong};\qquad [f]\mapsto f^\ast \E G,\]

is a well-defined bijection, and is natural in the sense that it is compatible with pullback along any morphism \(X'\rightarrow X\).

Proof

That \([f]\mapsto f^\ast \E G\) does not depend on the choice of representative for \([f]\) follows from Theorem 5 (Homotopy Invariance of Pullback). We briefly sketch that this is a bijection.

First, suppose a principal \(G\)-bundle \(P\) over \(X\) is given. Since \(X\) is paracompact Hausdorff, by [Topology] §Compactness and Paracompactness, ⁋Theorem 27 we may choose an open cover \(\{U_i\}\) trivializing \(P\) together with a locally finite partition of unity \(\{\rho_i\}\) subordinate to it. The trivialization over each \(U_i\) gives a \(G\)-equivariant map \(\psi_i:p^{-1}(U_i)\rightarrow G\), so taking \(\E G\) as the join from Theorem 7 (Milnor) and writing its points in the form \(\sum_i t_ig_i\), we have that

\[\widetilde{f}:P\rightarrow \E G,\qquad y\mapsto \sum_i \rho_i(p(y))\psi_i(y)\]

is a well-defined \(G\)-equivariant map. A \(G\)-equivariant map descends to a morphism \(f:X\rightarrow \B G\) between base spaces, and since \(\widetilde{f}\) is an isomorphism on each fiber, we obtain \(P\cong f^\ast\E G\).

Now to show injectivity, suppose for \(f_0,f_1:X\rightarrow \B G\) that \(f_0^\ast \E G\cong f_1^\ast \E G=:P\). We must show that \(f_0\) and \(f_1\) are homotopic. Each \(f_i\) yields a bundle map \(P\cong f_i^\ast \E G\rightarrow \E G\), that is, a \(G\)-equivariant bundle map \(\Phi_i:P\rightarrow \E G\) from \(P\) to the universal bundle. But since \(\E G\) is contractible, any two \(G\)-equivariant maps from a principal bundle \(P\) over a paracompact space to \(\E G\) are \(G\)-equivariantly homotopic; hence there exists a \(G\)-equivariant homotopy \(P\times[0,1]\rightarrow \E G\) joining \(\Phi_0\) and \(\Phi_1\), and this descends to the base to give a homotopy between \(f_0\) and \(f_1\), so \([f_0]=[f_1]\).

This theorem translates the geometric classification of principal \(G\)-bundles into purely homotopy-theoretic data \([X,\B G]\). Combined with Proposition 4, the classification of rank \(n\) real vector bundles becomes \([X,\B\GL(n;\mathbb{R})]\), and in the complex case \([X,\B\GL(n;\mathbb{C})]\); and indeed we shall soon see that these \(\B\GL(n; \mathbb{R})\) and \(\B\GL(n; \mathbb{C})\) are none other than the (real/complex) Grassmannians.

Lemma 9 The construction of classifying spaces is functorial in \(G\). Given a continuous group homomorphism \(\phi:G\rightarrow H\), changing the \(G\)-action on \(\E G\) to an \(H\)-action via \(\phi\) yields the associated bundle \(\E G\times_G H\), and the morphism classifying it induces \(\B\phi:\B G\rightarrow \B H\). This satisfies \(\B(\psi\circ\phi)\simeq \B\psi\circ \B\phi\), making \(G\mapsto \B G\) a functor on the homotopy category. For instance, the inclusion \(\Umat(n)\hookrightarrow\GL(n;\mathbb{C})\) induces \(\B\Umat(n)\rightarrow \B\GL(n;\mathbb{C})\), which will be used below.

Proof

The \(\B G\) from Theorem 7 (Milnor) is a CW complex, hence paracompact Hausdorff, so we may apply Theorem 8 (Classification Theorem) to principal bundles over \(\B G\). Now \(\B\phi\) is determined by \(\B\phi^\ast\E H\cong\E G\times_G H\), and since forming associated bundles commutes with pullback, for a continuous group homomorphism \(\psi:H\rightarrow K\) we obtain

\[(\B\psi\circ\B\phi)^\ast\E K\cong\B\phi^\ast\left(\E H\times_H K\right)\cong\left(\B\phi^\ast\E H\right)\times_H K\cong\left(\E G\times_G H\right)\times_H K\cong \E G\times_G K.\]

In the last term, the action of \(G\) on \(K\) is via \(\psi\circ\phi\), so this is exactly the bundle classified by \(\B(\psi\circ\phi)\); hence \(\B\psi\circ\B\phi\) and \(\B(\psi\circ\phi)\) classify the same principal \(K\)-bundle over \(\B G\). Then by the injectivity in Theorem 8 (Classification Theorem), they are homotopic.

Examples of Classifying Spaces

In practice, the most useful part of this post is not the existence result but rather how these classifying spaces are given explicitly. The simplest cases are as follows.

Example 10 Suppose \(G\) is a discrete group. Then any principal \(G\)-bundle over a base \(B\) has discrete fiber, so it becomes a covering space over \(B\). From this perspective, the right action of \(G\) becomes the Deck transformation, and since the Deck group acts transitively on the fiber, this covering space is a regular covering space.

Now apply this to the universal bundle \(\E G \rightarrow \B G\). Since \(\E G\) is contractible, it is the universal cover of \(\B G\), and since the \(\B G\) from Theorem 7 (Milnor) is a connected CW complex, the conditions required by covering space theory (path-connected, locally path-connected, and semi-locally simply connected) are all satisfied. Then, as we saw in §Covering Spaces, §§Fundamental Theorems of Covering Spaces, the Deck transformation group of this covering space is isomorphic to \(\pi_1(\B G)\); but we observed earlier that this Deck group must be \(G\), so \(\pi_1(\B G)\cong G\), and since \(\E G\) is contractible the universal cover of \(\B G\) is also contractible, hence \(\pi_n(\B G)=0\) for \(n\geq 2\). That is, \(\B G\) is an Eilenberg–MacLane space \(K(G,1)\).

For more concrete examples, let us consider the cases \(G=\mathbb{Z}/2\) and \(G=\mathbb{Z}\). First, for \(\mathbb{Z}/2\) we need a contractible space on which \(\mathbb{Z}/2\) acts freely, and the antipodal action on \(S^\infty\) satisfies both of these conditions. Then the orbit space of this action is \(\RP^\infty\). The case \(G=\mathbb{Z}\) can also be found in a familiar example: namely, the covering \(\mathbb{R}\rightarrow S^1\) introduced in §Covering Spaces, ⁋Definition 3 as the standard example of a covering space.

Now let us examine the classifying spaces of the groups we are actually interested in. Among non-discrete groups, the most basic example is \(G=S^1\), which is commonly thought of as the set of complex numbers of length \(1\), that is, the \(e^{2\pi it}\). Then \(S^1\) acts freely on \(\mathbb{C}^\infty\setminus\{0\}\) by scalar multiplication.

Now each \(\mathbb{C}^n\setminus 0\) can be deformation retracted onto the unit sphere

\[S^{2n-1}\subseteq\mathbb{C}^n\cong\mathbb{R}^{2n}\]

via radial deformation retraction, and the canonical inclusions

\[\mathbb{C}^n\hookrightarrow\mathbb{C}^{n+1}\hookrightarrow \mathbb{C}^{n+2}\hookrightarrow \cdots\]

induce inclusions \(S^{2n-1}\hookrightarrow S^{2n+1}\) sending the former unit sphere to the equator of the next, as one can verify. Thus, viewing \(\mathbb{C}^\infty\setminus \{0\}\) as the colimit \(\varinjlim (\mathbb{C}^n\setminus \{0\})\), this deformation retracts to the colimit \(\varinjlim S^{2n-1}\), which is a cofinal subsequence of the inclusions

\[S^1\subseteq S^2\subseteq S^3\cdots\]

appearing in the definition of \(S^\infty\), so the result is the same as \(S^\infty\). On the other hand, since scalar multiplication by \(S^1\) preserves the norm, this action restricts to a free action on the unit sphere \(S^\infty\subseteq\mathbb{C}^\infty\setminus\{0\}\). That is, taking \(\E S^1=S^\infty\) gives a contractible space on which \(S^1\) acts freely, and since each complex line in \(\mathbb{C}^\infty\) meets \(S^\infty\) in exactly one \(S^1\)-orbit (namely, the unit circle in that line), the orbit space is the complex projective space

\[\B S^1=S^\infty/S^1=\CP^\infty.\]

To see what this means in the language of vector bundles, let us return to the associated bundle of Definition 3. Since \(S^1\) acts on \(\mathbb{C}\) by scalar multiplication, for any principal \(S^1\)-bundle \(P\rightarrow X\) the associated bundle

\[P\times_{S^1}\mathbb{C}\rightarrow X\]

is defined. As we saw earlier, this is a bundle with fiber \(\mathbb{C}\) over the same open cover as \(P\) and transition functions given by the action of \(g_{ij}\) on \(\mathbb{C}\); since scalar multiplication is \(\mathbb{C}\)-linear, these transitions are linear automorphisms given by elements of \(S^1\subseteq\mathbb{C}^\times=\GL(1;\mathbb{C})\), and hence \(P\times_{S^1}\mathbb{C}\) is a complex line bundle. That is, a principal \(S^1\)-bundle naturally becomes a line bundle simply by attaching \(\mathbb{C}\). Conversely, given a line bundle \(L\rightarrow X\), paracompactness allows us to choose a Hermitian metric, and the sphere bundle \(S(L)\subseteq L\) consisting of unit vectors in each fiber becomes a principal \(S^1\)-bundle under scalar multiplication by \(S^1\). By the same argument as in Proposition 4, where ordered bases were replaced by unit vectors, one can verify that these two constructions are inverse to each other, and this is why the structure group of a line bundle can be reduced from \(\GL(1;\mathbb{C})\) to \(S^1=\Umat(1)\).

Applying this explicitly to the universal bundle \(\E S^1=S^\infty\rightarrow\CP^\infty\) yields the line bundle

\[S^\infty\times_{S^1}\mathbb{C}\longrightarrow\CP^\infty.\]

Examining the fiber over a point \([\ell]\in\CP^\infty\), the equivalence class \([e,z]\) equals \(ze\in\ell\) where the unit vector \(e\) determines the line \(\ell=\mathbb{C}e\), so this is the tautological line bundle \(\gamma\) having each line as its own fiber. That is, the reason \(\gamma\) was the universal family of complex line bundles in §Characteristic Classes of Vector Bundles, ⁋Example 8 is that attaching \(\mathbb{C}\) to the universal principal \(S^1\)-bundle yields exactly \(\gamma\), and conversely the \(S^\infty\) appearing there as the sphere bundle of \(\gamma\) is precisely \(\E S^1\).

Example 11 (Classifying Spaces of Linear Groups) The above discussion generalizes to arbitrary rank \(n\) bundles. First, observe that if a continuous representation

\[G\rightarrow\GL(n;\mathbb{C})\]

of a topological group \(G\) is given, then for any principal \(G\)-bundle \(P\) the associated bundle \(P\times_G\mathbb{C}^n\) becomes a rank \(n\) complex vector bundle. Since a line bundle was obtained by attaching the standard representation \(\mathbb{C}\) to a principal \(\Umat(1)=S^1\)-bundle, it is natural to expect that a rank \(n\) complex vector bundle is obtained by attaching the standard representation \(\mathbb{C}^n\) to a principal \(\Umat(n)\)-bundle.

What is needed for this is a universal principal \(\Umat(n)\)-bundle, which is given by the complex Stiefel manifold

\[V_n(\mathbb{C}^\infty)=\varinjlim_k V_n(\mathbb{C}^k),\]

the space of all orthonormal \(n\)-frames in \(\mathbb{C}^\infty\), with \(\Umat(n)\) acting on the right by matrix multiplication and orbit space \(\Gr(n,\mathbb{C}^\infty)\). An orthonormal \(1\)-frame is just a unit vector, so for \(n=1\) this is exactly \(\E S^1=S^\infty\rightarrow\CP^\infty\) from the main text, and for general \(n\) the same argument as for \(S^\infty\) shows that \(V_n(\mathbb{C}^\infty)\) deformation retracts to a point, hence is contractible, so this principal bundle is universal.

Now attaching the canonical representation \(\mathbb{C}^n\) yields the associated bundle

\[V_n(\mathbb{C}^\infty)\times_{\Umat(n)}\mathbb{C}^n\longrightarrow\Gr(n,\mathbb{C}^\infty).\]

Examining the fiber over a point \([V]\), for \(z=(z_1,\ldots,z_n)\in\mathbb{C}^n\) the equivalence class \([(e_1,\ldots,e_n),z]\) equals \(z_1e_1+\cdots+z_ne_n\in V\), the element of the subspace \(V\) spanned by the frame; thus, just as for line bundles, this is the tautological \(n\)-plane bundle \(\gamma^n\) having each subspace as its own fiber, and conversely collecting all orthonormal frames of each fiber of \(\gamma^n\) recovers \(V_n(\mathbb{C}^\infty)\), again as in the line bundle case. That is,

\[\B\Umat(n)=\Gr(n,\mathbb{C}^\infty)\]

and the universal bundle over it is the tautological \(n\)-plane bundle.

On the other hand, Proposition 4 (more precisely, the complex version of that proposition) assigns to rank \(n\) complex vector bundles a process that strictly speaking uses associated bundles via principal \(\GL(n;\mathbb{C})\)-bundles. That is, for the above computation to lead to the classification of arbitrary complex vector bundles, \(\B\GL(n;\mathbb{C})\) and \(\B\Umat(n)\) must be the same, and indeed they are. This follows from [Linear Algebra] §Complex Inner Product Spaces, ⁋Proposition 7 (QR decomposition): any element of \(\GL(n;\mathbb{C})\) decomposes uniquely as a product of a unitary matrix and an upper-triangular matrix with positive diagonal entries, and one can show that this decomposition is continuous. Contracting the upper-triangular component toward the identity then gives a deformation retract of \(\GL(n;\mathbb{C})\) onto \(\Umat(n)\). That is, the inclusion \(\Umat(n)\hookrightarrow\GL(n;\mathbb{C})\) is a homotopy equivalence, and by the functoriality of classifying spaces Lemma 9,

\[\B\GL(n;\mathbb{C})\simeq \B\Umat(n)=\Gr(n,\mathbb{C}^\infty).\]

Repeating the entire story for the real Stiefel manifold \(V_n(\mathbb{R}^\infty)\) of orthonormal \(n\)-frames in \(\mathbb{R}^\infty\), with \(\Omat(n)\), and Gram–Schmidt orthogonalization, we obtain

\[\B\GL(n;\mathbb{R})\simeq \B\Omat(n)=\Gr(n,\mathbb{R}^\infty).\]

Cohomology of Classifying Spaces

By Theorem 8 (Classification Theorem), characteristic classes of bundles with structure group \(G\) are precisely cohomology classes of \(\B G\) pulled back via the classifying map. Thus characteristic class theory is the same as computing the cohomology ring of \(\B G\), and we summarize this for the most basic groups.

The starting point is the cohomology ring of complex projective space. In §Characteristic Classes of Vector Bundles, ⁋Example 8 we saw that

\[H^\bullet(\CP^\infty;\mathbb{Z})=\mathbb{Z}[t],\qquad \lvert t\rvert=2\]

and the generator \(t\) was the first Chern class of the tautological line bundle. Since we showed above that \(\B S^1=\CP^\infty\), this means

\[H^\bullet(\B S^1;\mathbb{Z})=\mathbb{Z}[t],\qquad \lvert t\rvert=2.\]

The case of the torus follows from the cohomology of product spaces.

Corollary 12 For an \(n\)-dimensional torus \(T=(S^1)^n\),

\[H^\bullet(\B T;\mathbb{Z})=\mathbb{Z}[t_1,\ldots,t_n],\qquad \lvert t_i\rvert=2\]

is a polynomial ring generated by \(n\) generators of degree \(2\). Moreover, the degree \(2\) part \(H^2(\B T;\mathbb{Z})\) is canonically isomorphic to \(\Hom(T,S^1)\).

Proof

Since \(\B T=(\CP^\infty)^n\), let \(\pi_i:\B T\rightarrow\CP^\infty\) be the projection onto the \(i\)-th factor. From the computation \(\B S^1=\CP^\infty\) in the previous section, the cohomology of each factor \(H^\bullet(\CP^\infty;\mathbb{Z})=\mathbb{Z}[t]\) is a free abelian group of finite rank in each degree, so by the Künneth formula for cohomology given by §Cohomology, ⁋Corollary 10 (Künneth) and §Cohomology, ⁋Theorem 5 (Universal coefficient theorem for cohomology, general version), the \(\Tor\) and \(\Ext\) terms all vanish. Thus the cross product is an isomorphism in each degree, and by §Cup Product, ⁋Proposition 3 this is a graded algebra homomorphism, so by induction on the number of factors we obtain the ring isomorphism

\[H^\bullet(\B T;\mathbb{Z})\cong\bigotimes_{i=1}^n \mathbb{Z}[t_i]=\mathbb{Z}[t_1,\ldots,t_n]\]

where the generator \(t_i\) is the pullback of the generator \(t\) of the \(i\)-th factor via \(\pi_i\), that is, \(t_i=\pi_i^\ast t\).

Now consider the degree \(2\) part. A character \(\rchi:T\rightarrow S^1\) induces

\[\B\rchi:\B T\rightarrow \B S^1=\CP^\infty\]

by functoriality, so the pullbacks \((\B\rchi)^\ast t\in H^2(\B T;\mathbb{Z})\) of the generator \(t\) are determined. One can verify that this correspondence \(\rchi\mapsto(\B\rchi)^\ast t\) gives a homomorphism \(\Hom(T,S^1)\rightarrow H^2(\B T;\mathbb{Z})\), and the crucial point is that, as we saw above, the \(i\)-th coordinate projection \(\pr_i:T\rightarrow S^1\) maps exactly to \(t_i\). That is, \(\B\pr_i\) is exactly the \(i\)-th projection \(\pi_i\), and therefore

\[(\B\pr_i)^\ast t=\pi_i^\ast t=t_i.\]

Thus the standard basis \(\{\pr_1,\ldots,\pr_n\}\) of \(\Hom(T,S^1)\cong\mathbb{Z}^n\) maps to the basis \(\{t_1,\ldots,t_n\}\) of \(H^2(\B T;\mathbb{Z})=\bigoplus_i\mathbb{Z}t_i\), so this correspondence is an isomorphism.

This isomorphism allows us to read polynomials on the character lattice as cohomology classes of \(\B T\), and it becomes central when dealing with invariants of spaces with a torus action. The case of the unitary group requires a computation one step further, but we have already seen the result in the previous post.

Proposition 13 For the unitary group \(\Umat(n)\),

\[H^\bullet(\B\Umat(n);\mathbb{Z})=\mathbb{Z}[c_1,\ldots,c_n],\qquad \lvert c_i\rvert=2i\]

is the polynomial ring generated by the Chern classes \(c_i\) of the universal complex bundle.

Proof

We have \(\B\Umat(n)=\Gr(n,\mathbb{C}^\infty)\), and the fact that its cohomology ring is the polynomial ring generated by the Chern classes of the universal bundle,

\[H^\bullet(\Gr(n,\mathbb{C}^\infty);\mathbb{Z})=\mathbb{Z}[c_1,\ldots,c_n],\]

was already discussed after §Characteristic Classes of Vector Bundles, ⁋Example 8. Thus it suffices to show that the generators are Chern classes and that \(\lvert c_i\rvert=2i\).

This computation is essentially the same as Corollary 12, and the key point is, just as before, the map \(\B T\rightarrow\B\Umat(n)\) obtained by including the maximal torus \(T=(S^1)^n\subseteq\Umat(n)\) as diagonal matrices. Restricting the canonical representation \(\mathbb{C}^n\) of \(\Umat(n)\) to \(T\) splits along the coordinate axes as

\[\mathbb{C}^n=L_1\oplus\cdots\oplus L_n,\]

and \(T\) acts on the \(i\)-th line \(L_i\) exactly by the character \(\pr_i\). Therefore the pullback of the universal bundle \(E\) to \(\B T\) is the sum \(\bigoplus_i\mathcal{L}_i\) of the line bundles associated to each character, and its \(i\)-th summand is precisely the line bundle for which we already computed \(c_1(\mathcal{L}_i)=(\B\pr_i)^\ast t=t_i\) in Corollary 12. Applying the Whitney formula gives

\[c(E)\vert_{\B T}=\prod_{i=1}^n(1+t_i);\qquad c_i\vert_{\B T}=e_i(t_1,\ldots,t_n).\]

Here \(e_i\) is the \(i\)-th elementary symmetric polynomial, and since \(\lvert t_i\rvert=2\), we have \(\lvert c_i\rvert=2i\). What remains is that \(H^\bullet(\B\Umat(n);\mathbb{Z})\rightarrow H^\bullet(\B T;\mathbb{Z})=\mathbb{Z}[t_1,\ldots,t_n]\) is injective and its image is the invariant ring \(\mathbb{Z}[t_1,\ldots,t_n]^{S_n}\) under the Weyl group \(S_n\); since symmetric polynomials with integer coefficients are freely generated by elementary symmetric polynomials, we have \(\mathbb{Z}[t_1,\ldots,t_n]^{S_n}=\mathbb{Z}[e_1,\ldots,e_n]=\mathbb{Z}[c_1,\ldots,c_n]\), and hence the cohomology of \(\B\Umat(n)\) is exactly the \(S_n\)-symmetric part of the polynomial ring in Corollary 12. We refer to [MS] for the detailed computation.

Thus, since the cohomology of \(\B\Umat(n)\) consists entirely of polynomials in the Chern classes, every characteristic class of a complex vector bundle is a polynomial in the Chern classes. In the same way, \(H^\bullet(\B\Omat(n);\mathbb{Z}/2)=\mathbb{Z}/2[w_1,\ldots,w_n]\) gives the Stiefel–Whitney classes, and for oriented bundles \(\B\SO(n)\) gives the Euler class. When dealing with spaces equipped with a \(G\)-action instead of a single space \(X\), \(\B G\) and the homotopy quotient over it form the foundation of equivariant cohomology, taking this cohomology as its base.

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References

[Hat] A. Hatcher, Vector Bundles and K-Theory, online notes, 2017.
[MS] J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974.