Section 17.1 Classifying Spaces for Vector Bundles

Recall (Example 16.5) that the Grassmannian \(\mathrm{Gr}_k(\mathbb{C}^n)\) of complex \(k\)-planes in \(\mathbb{C}^n\) carries a tautological rank-\(k\) complex vector bundle. Allowing \(n \to \infty\) produces the infinite Grassmannian

\begin{equation*} \mathrm{Gr}_k(\mathbb{C}^\infty) \;:=\; \mathop{\mathrm{colim}}_{n \to \infty} \mathrm{Gr}_k(\mathbb{C}^n), \end{equation*}

endowed with the colimit topology. Here \(\mathbb{C}^\infty = \bigcup_n \mathbb{C}^n\text{,}\) so a point of \(\mathrm{Gr}_k(\mathbb{C}^\infty)\) is a \(k\)-dimensional subspace of \(\mathbb{C}^\infty\text{.}\) Its tautological bundle is

\begin{equation*} \gamma_k \;=\; \bigl\{\,(W, v) \in \mathrm{Gr}_k(\mathbb{C}^\infty) \times \mathbb{C}^\infty \;\big|\; v \in W \,\bigr\}, \end{equation*}

topologised as the colimit of the tautological bundles over the \(\mathrm{Gr}_k(\mathbb{C}^n)\text{.}\) The same construction over \(\mathbb{R}\) produces \(\mathrm{Gr}_k(\mathbb{R}^\infty)\) together with a real tautological bundle.

Definition 17.1.

The classifying space for complex rank-\(k\) vector bundles is

\begin{equation*} BU(k) \;:=\; \mathrm{Gr}_k(\mathbb{C}^\infty), \end{equation*}

and the classifying space for real rank-\(k\) vector bundles is

\begin{equation*} BO(k) \;:=\; \mathrm{Gr}_k(\mathbb{R}^\infty). \end{equation*}

The tautological bundle \(\gamma_k \to BU(k)\) (resp. \(\to BO(k)\)) is called the universal bundle.

Theorem 17.2. Classification of vector bundles.

Let \(X\) be a CW complex. Pullback of the universal bundle gives a natural bijection

\begin{equation*} [X,\, BU(k)] \;\xrightarrow{\;\cong\;}\; \mathrm{Vect}_k^{\mathbb{C}}(X), \qquad [f] \;\longmapsto\; [f^*\gamma_k]. \end{equation*}

The analogous statement holds in the real case with \(BO(k)\) and \(\mathrm{Vect}_k^{\mathbb{R}}(X)\text{.}\)

The map is well defined by homotopy invariance (Theorem 16.7), since CW complexes are paracompact (Hatcher, Vector Bundles and K-Theory, Prop. 1.20). We prove the complex case with the machinery of Chapter 15. First we check that \(\mathrm{Vect}_k^{\mathbb{C}}\) satisfies the hypotheses of Brown’s theorem. The theorem then produces a classifying space abstractly, and we identify it with the Grassmannian by comparing what the two spaces do on spheres. For the rest of this section we write \(\mathrm{Vect}_k\) for \(\mathrm{Vect}_k^{\mathbb{C}}\) and assume \(k \geq 1\text{,}\) since for \(k = 0\) both sides of the theorem are a single point.

Lemma 17.3.

The functor \(\mathrm{Vect}_k \colon \mathrm{Ho}(\mathbf{CW}_{*,\mathrm{conn}})^{\mathrm{op}} \to \mathbf{Set}_*\text{,}\) pointed by the trivial bundle \(\varepsilon^k\text{,}\) satisfies (B1) and (B2).

Proof.

Sketch. Based homotopic maps are in particular homotopic, so \(\mathrm{Vect}_k\) is a functor on the based homotopy category. Both conditions come down to gluing bundles, and the only subtle point is local triviality where the pieces meet.

For (B2) we treat the case where \(X = B \cup C\) is a CW complex with subcomplexes \(B\) and \(C\text{,}\) and \(A = B \cap C\text{.}\) This is the form in which the condition enters the proof of Brown’s theorem (Hatcher, Theorem 4E.2), and the general case reduces to it with mapping cylinders as in Proposition 15.9. Let \(E_B \to B\) and \(E_C \to C\) be bundles with an isomorphism \(\phi \colon E_B|_A \to E_C|_A\text{,}\) and let \(E\) be the quotient of \(E_B \sqcup E_C\) by \(v \sim \phi(v)\) for \(v \in E_B|_A\text{.}\) Then \(E\) restricts to \(E_B\) and \(E_C\text{,}\) and it is locally trivial away from \(A\text{.}\) Near a point \(a \in A\text{,}\) choose trivializations of \(E_B\) and \(E_C\) over neighbourhoods of \(a\) in \(B\) and in \(C\text{.}\) Over \(A\) they differ, via \(\phi\text{,}\) by a map \(\tau\) to \(\mathrm{GL}(k, \mathbb{C})\) (Exercise 16.23). By the Tietze extension theorem, applied to the entries of \(\tau\text{,}\) it extends to a matrix-valued map on a neighbourhood of \(a\) in \(B\text{,}\) which is still invertible near \(a\) because \(\mathrm{GL}(k, \mathbb{C})\) is open in the space of matrices. Correcting the trivialization of \(E_B\) by this extension makes the two trivializations agree on \(A\text{,}\) so they glue to a trivialization of \(E\) near \(a\text{.}\)

For (B1), a bundle on \(\bigvee_\alpha X_\alpha\) is obtained in the same way from bundles \(E_\alpha \to X_\alpha\) together with identifications of their fibres over the basepoint \(x_0\text{.}\) So the map in (B1) is surjective. For injectivity, let \(E\) and \(E'\) be bundles on the wedge with isomorphisms \(\phi_\alpha \colon E|_{X_\alpha} \to E'|_{X_\alpha}\text{.}\) These need not agree on the fibre over \(x_0\text{,}\) so we correct them. Fix an index \(\alpha_0\text{.}\) We compose each \(\phi_\alpha\) with an automorphism of \(E'|_{X_\alpha}\) that equals \(\phi_{\alpha_0}(x_0)\,\phi_\alpha(x_0)^{-1}\) on the fibre over \(x_0\text{.}\) To build it, choose a trivialization of \(E'\) over a neighbourhood \(U\) of \(x_0\) in \(X_\alpha\text{,}\) a function \(\lambda \colon X_\alpha \to [0,1]\) with \(\lambda(x_0) = 1\) and support in \(U\text{,}\) and a path \(\omega\) in \(\mathrm{GL}(k, \mathbb{C})\) from the identity to the required matrix. This path exists because \(\mathrm{GL}(k, \mathbb{C})\) is path-connected. The automorphism acts by \(\omega(\lambda(x))\) on the fibre over \(x \in U\) and by the identity outside \(U\text{.}\) The corrected isomorphisms agree over \(x_0\) and glue to an isomorphism \(E \cong E'\text{.}\)

By Lemma 17.3 and Brown’s theorem (Theorem 15.10) there are a pointed connected CW complex \(B_k\) and a bundle \(u_k \to B_k\) such that

\begin{equation*} [Y, B_k]_* \;\xrightarrow{\;\cong\;}\; \mathrm{Vect}_k(Y), \qquad [f] \;\longmapsto\; [f^*u_k], \end{equation*}

for every pointed connected CW complex \(Y\text{.}\) Taking \(Y = S^n\) computes the homotopy groups of \(B_k\text{,}\)

\begin{equation*} \pi_n(B_k) \;\cong\; \mathrm{Vect}_k(S^n). \end{equation*}

So the homotopy groups of a classifying space are the bundles on spheres. In particular \(\pi_1(B_k) \cong \mathrm{Vect}_k(S^1)\) is trivial, since every complex bundle on \(S^1\) is trivial (Hatcher, Example 1.12, and Exercise 16.26). The fibres of the forgetful map \([Y, B_k]_* \to [Y, B_k]\) are the orbits of \(\pi_1(B_k)\text{,}\) so based and unbased homotopy classes agree, and \([Y, B_k] \cong \mathrm{Vect}_k(Y)\text{.}\)

Brown’s theorem tells us that \(B_k\) exists, but not what it is. To identify it with the Grassmannian it suffices, by Whitehead’s theorem, to compare homotopy groups. On the side of \(B_k\) these are bundles on spheres, so we need the following.

Lemma 17.4.

For \(n \geq 1\) the map \(\pi_n(BU(k)) \to \mathrm{Vect}_k(S^n)\text{,}\) \([f] \mapsto [f^*\gamma_k]\text{,}\) is a bijection.

Proof.

Sketch. We show that both sides are \(\pi_{n-1}(U(k))\text{.}\) Let \(V_k(\mathbb{C}^\infty)\) be the Stiefel manifold of orthonormal \(k\)-frames in \(\mathbb{C}^\infty\text{,}\) topologised as a colimit like the Grassmannian. Sending a frame to the plane it spans gives a fibre bundle

\begin{equation*} U(k) \longrightarrow V_k(\mathbb{C}^\infty) \longrightarrow BU(k). \end{equation*}

Indeed, over the set of planes \(W\) that project isomorphically onto a fixed plane \(W_0\) (Example 16.5), applying Gram–Schmidt to the preimage in \(W\) of a fixed basis of \(W_0\) gives a continuous choice of frame, and every other frame of \(W\) differs from it by a unique element of \(U(k)\text{.}\) The Stiefel manifold is contractible. Let \(L(e_i) = e_{2i-1}\text{.}\) The homotopy \((1-t)v + tL(v)\) applied to each vector of a frame, followed by Gram–Schmidt, moves every frame into the odd coordinates. From there the straight line to the fixed frame \((e_2, e_4, \dots, e_{2k})\text{,}\) followed by Gram–Schmidt, contracts it. Since \(BU(k)\) is paracompact, Theorem 10.14 and the long exact sequence (Theorem 10.9) give a bijection \(\partial \colon \pi_n(BU(k)) \to \pi_{n-1}(U(k))\text{.}\)

On the other side, a bundle over \(S^n\) is trivial over the two hemispheres \(D^n_\pm\text{,}\) which are contractible (Corollary 16.8). Over the equator the two trivializations differ by a clutching function \(S^{n-1} \to \mathrm{GL}(k, \mathbb{C})\) (Exercise 16.23), and this gives a bijection \(\mathrm{Vect}_k(S^n) \cong [S^{n-1}, \mathrm{GL}(k, \mathbb{C})]\) (Hatcher, Prop. 1.11). Gram–Schmidt deformation retracts \(\mathrm{GL}(k, \mathbb{C})\) onto \(U(k)\text{.}\) Since \(U(k)\) is a path-connected topological group, unbased and based homotopy classes of maps into it agree, so \(\mathrm{Vect}_k(S^n) \cong \pi_{n-1}(U(k))\text{.}\)

The two identifications match. For \(f \colon S^n \to BU(k)\text{,}\) lift \(f\) over each hemisphere to \(V_k(\mathbb{C}^\infty)\text{.}\) Such a lift is a unitary frame of \(f^*\gamma_k\) over that hemisphere, that is, a trivialization. Over the equator the two lifts differ by a map \(S^{n-1} \to U(k)\text{.}\) This map is the clutching function of \(f^*\gamma_k\text{,}\) and unwinding the definition of the boundary map shows that it represents \(\partial[f]\text{.}\)

Proof.

Proof of Theorem 17.2. Both sides take disjoint unions to products, so we may assume that \(X\) is connected and choose a basepoint. The Grassmannian \(BU(k)\) is a connected CW complex (Hatcher, §1.2, constructs its cells). Applying the bijection \([BU(k), B_k]_* \cong \mathrm{Vect}_k(BU(k))\) to \(\gamma_k\) gives a based map \(c \colon BU(k) \to B_k\) with \(c^*u_k \cong \gamma_k\text{.}\) For every pointed connected CW complex \(Y\text{,}\) the composite

\begin{equation*} [Y, BU(k)]_* \xrightarrow{\;c_*\;} [Y, B_k]_* \xrightarrow{\;\cong\;} \mathrm{Vect}_k(Y) \end{equation*}

sends \([f]\) to \([f^*c^*u_k] = [f^*\gamma_k]\text{.}\) For \(Y = S^n\) the composite is a bijection by Lemma 17.4, so \(c_* \colon \pi_n(BU(k)) \to \pi_n(B_k)\) is a bijection for every \(n \geq 1\text{,}\) and \(c\) is a weak equivalence. By Whitehead’s theorem (Theorem 8.11) it is a homotopy equivalence, so \(c_*\) is a bijection for every \(Y\text{.}\) Hence \([f] \mapsto [f^*\gamma_k]\) is a bijection \([Y, BU(k)]_* \cong \mathrm{Vect}_k(Y)\text{.}\) Finally \(BU(k) \simeq B_k\) is simply connected, so based and unbased classes agree as before.

Remark 17.5.

Hatcher (Vector Bundles and K-Theory, Theorem 1.16) gives a direct proof that works for every paracompact \(X\text{.}\) The key observation is that a map \(f \colon X \to BU(k)\) with \(f^*\gamma_k \cong E\) is the same as a map \(g \colon E \to \mathbb{C}^\infty\) that is a linear injection on each fibre, with \(f(x) = g(E_x)\text{.}\) Such a \(g\) is built from local trivializations and a partition of unity. Two such maps are homotopic through maps of the same kind, by moving one into the odd coordinates, the other into the even coordinates, and using the straight-line homotopy. This is the argument that showed above that the Stiefel manifold is contractible, applied one fibre at a time. The direct proof also handles real bundles. The approach through Brown’s theorem needs care there, since \(\mathrm{GL}(k, \mathbb{R})\) is not path-connected and the argument for (B1) breaks down. One way around this is to apply Brown’s theorem to oriented bundles instead.

Remark 17.6.

The notation \(BU(k)\text{,}\) \(BO(k)\) comes from the theory of classifying spaces of topological groups. For any topological group \(G\) there is a space \(BG\) classifying principal \(G\)-bundles over paracompact spaces, and one has \(BU(k) \simeq B(U(k))\) for \(U(k)\) the unitary group, and \(BO(k) \simeq B(O(k))\) for the orthogonal group. The identification with the Grassmannian uses the fact that, over a paracompact base, a rank-\(k\) complex vector bundle is the same data as a principal \(U(k)\)-bundle. Indeed, after choosing a Hermitian metric (which exists by a partition of unity argument), the bundle of unitary frames is a principal \(U(k)\)-bundle.

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