In this section, we introduce the classical theory of complex vector bundles and isolate the input needed later: morphisms, pullbacks, direct sums, tensor products, and homotopy invariance.
Definition 9.1.1. Let \(X\) be a topological space. A finite-rank complex vector bundle over \(X\) is a continuous map \(p\colon E\to X\), together with a complex vector space structure on every fiber \(E_x:=p^{-1}(x)\), satisfying the following local triviality condition: every point \(x\in X\) admits an open neighborhood \(U\), an integer \(k\geq 0\), and a homeomorphism \[ \phi \colon U\times \C ^k\xrightarrow {\cong }p^{-1}(U) \] over \(U\) whose restriction \(\{y\}\times \C ^k\to E_y\) is complex-linear for every \(y\in U\). We will usually simply call this a complex vector bundle.
The rank function \[ \rk _E\colon X\longrightarrow \N , \qquad x\longmapsto \dim _{\C }(E_x), \] is locally constant. We say that \(E\) has rank \(k\) if this function is constant with value \(k\). More generally, each subspace \(X_k:=\rk _E^{-1}(k)\) is clopen in \(X\), and the restriction of \(E\) to \(X_k\) has rank \(k\).
Given another complex vector bundle \(q\colon E'\to X\), a morphism of vector bundles \(E\to E'\) is a continuous map \(f\colon E\to E'\) over \(X\) whose restriction \(E_x\to E'_x\) is complex-linear for every \(x\in X\). We write \(\pi _0\Vect (X)\) for the set of isomorphism classes of finite-rank complex vector bundles over \(X\), and \(\pi _0\Vect _k(X)\) for the subset represented by bundles of rank \(k\).
Let \(X\) be a topological space, and let \(p_1\colon E_1 \to X\) and \(p_2\colon E_2 \to X\) be two complex vector bundles over \(X\). Their direct sum is defined as the fiber product \[ E_1 \oplus E_2 \quad := \quad E_1 \times _X E_2 \quad = \quad \{(e_1,e_2) \mid p_1(e_1) = p_2(e_2)\}, \] equipped on each fiber with the usual direct-sum vector space structure. We leave it to the reader to check that this is again a vector bundle; see Chapterexercise 9.3. The trivial rank-zero bundle \(\ul {0} := X \times \C ^0\) serves as a unit for direct sum up to the evident isomorphisms, and the usual associativity and symmetry isomorphisms show that direct sum induces an abelian monoid structure on \(\pi _0\Vect (X)\).
We will also use the tensor product of two vector bundles \(E_1\) and \(E_2\). This is the vector bundle \(E_1 \otimes E_2 \to X\) whose fiber over \(x \in X\) is \(E_{1,x} \otimes _{\C } E_{2,x}\); local trivializations of \(E_1\) and \(E_2\) induce local trivializations of \(E_1 \otimes E_2\). Pullback along maps of spaces is compatible with direct sums and tensor products, and tensor product distributes over direct sums up to isomorphism. Thus tensor product induces a multiplication on \(\pi _0\Vect (X)\), compatible with its direct-sum monoid structure.
9.1.1 Homotopy invariance of vector bundles
We will use the following classical homotopy invariance theorem for vector bundles over paracompact spaces.
Definition 9.1.2. A topological space \(X\) is called paracompact if for every open cover \(\{U_{i}\}_{i \in I}\) of \(X\) there exists a partition of unity subordinate to this cover, i.e. a collection of continuous maps \(\rho _{\alpha }\colon X \to [0,1]\) satisfying the following three conditions:
- (1)
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The family \((\rho _\alpha )_{\alpha }\) is locally finite: every point of \(X\) has a neighborhood on which all but finitely many \(\rho _\alpha \) vanish;
- (2)
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The resulting sum satisfies \(\sum _{\alpha } \rho _{\alpha }(x) = 1\) for all \(x\);
- (3)
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For each \(\alpha \), the support \(\supp (\rho _\alpha ):=\overline {\rho _{\alpha }^{-1}((0,1])}\) is contained in one of the open sets \(U_i\).
For Hausdorff spaces, this is equivalent to the usual definition in terms of locally finite refinements of open covers.
Example 9.1.3. The following classes of topological spaces are paracompact:
- Every compact Hausdorff space is paracompact [Hatcher (2003), Proposition 1.18];
- Every direct limit, with the colimit topology, of an increasing sequence \(X_1 \subseteq X_2 \subseteq \dots \) of compact Hausdorff spaces is paracompact [Hatcher (2003), Proposition 1.19];
- Every CW-complex is paracompact [Hatcher (2003), Proposition 1.20].
Theorem 9.1.4 (Homotopy invariance of vector bundles, Hatcher (2003), Theorem 1.6 and Corollary 1.8). Let \(p\colon E \to X\) be a complex vector bundle and let \(f_0,f_1\colon Y \to X\) be homotopic continuous maps. If \(Y\) is paracompact Hausdorff, then the pullback bundles \(f_0^*E\) and \(f_1^*E\) are isomorphic.
In particular, every homotopy equivalence \(f\colon X \to Y\) between paracompact Hausdorff spaces induces a bijection \[ f^*\colon \pi _0\Vect (Y) \xrightarrow {\cong } \pi _0\Vect (X). \]
Remark 9.1.5. The proof uses the usual partition-of-unity argument: a bundle on \(Y \times [0,1]\) is locally constant in the interval direction after subdividing \([0,1]\), and paracompactness lets one patch the resulting local isomorphisms. We will use only the conclusion above.
9.1.2 Clutching functions
Let \(X = U_+ \cup U_-\) be a topological space written as the union of two open subspaces, and let \(E \to X\) be a rank \(n\) complex vector bundle which is trivial over both \(U_+\) and \(U_-\). Choosing trivializations \[ E\vert _{U_+} \cong U_+ \times \C ^n \qquadtext {and}\qquad E\vert _{U_-} \cong U_- \times \C ^n \] identifies the gluing of these two trivial bundles along \(U_+ \cap U_-\) with a continuous map \[ g\colon U_+ \cap U_- \to \GL _n(\C ), \] called the clutching function of \(E\) with respect to the chosen trivializations. Conversely, any such map \(g\) defines a vector bundle by the quotient \[ E_g := (U_+ \times \C ^n) \sqcup (U_- \times \C ^n) \,/\, (x,v)_+ \sim (x,g(x)v)_- \] for \(x \in U_+ \cap U_-\). Thus clutching functions give a concrete way of building vector bundles by gluing trivial bundles.
For spheres this construction gives an elementary classification. Let \(m \geq 1\), and let \(U_+\) and \(U_-\) be slightly enlarged upper and lower hemispheres in \(S^m\). Then \(U_+\) and \(U_-\) are contractible paracompact Hausdorff spaces, and their intersection deformation retracts onto the equator. Informally, we are using the decomposition \[ S^m = D^m_+ \cup _{S^{m-1}} D^m_-. \] By homotopy invariance, every vector bundle over either \(U_+\) or \(U_-\) is trivial, so a rank \(n\) complex vector bundle over \(S^m\) is obtained from a clutching function \[ g\colon S^{m-1} \to \GL _n(\C ). \] The clutching construction induces a bijection \[ \pi _0\Vect _n(S^m) \cong [S^{m-1},\GL _n(\C )] \] with the set of free homotopy classes [Hatcher (2003), Proposition 1.11]. Since \(\GL _n(\C )\) is a connected topological group, based and free homotopy classes from a sphere agree, and hence this set is \(\pi _{m-1}(\GL _n(\C ))\). The required connectedness is verified in Exercise 9.1.6; for \(m=1\), the last statement reads \(\pi _0\Vect _n(S^1)\cong \pi _0\GL _n(\C )\).
Exercise 9.1.6 (Vector bundles over the circle). Show that the topological group \(\GL _n(\C )\) is connected. Deduce that every rank \(n\) complex vector bundle over \(S^1\) is trivial, so that \[ \pi _0\Vect (S^1) \cong \N . \]
Exercise 9.1.7 (Vector bundles over the two-sphere). Compute that \(\pi _1(\GL _1(\C ))\cong \Z \), with \(m\in \Z \) represented by the clutching function \(g_m\colon S^1\to \C ^\times \), \(z\mapsto z^m\), and construct an explicit representative \(L_m\) for the corresponding line bundle over \(S^2\). Next, show that the inclusion \(U(n) \hookrightarrow \GL _n(\C )\) is a homotopy equivalence. Using the fiber sequence \[ U(n-1) \hookrightarrow U(n) \twoheadrightarrow S^{2n-1}, \] deduce that \(\pi _1(\GL _n(\C )) \cong \Z \) for all \(n \geq 1\). Conclude that \(\pi _0\Vect _k(S^2) \cong \Z \) for all \(k \geq 1\), with every rank \(k\) complex vector bundle over \(S^2\) isomorphic to \[ L_m \oplus \ul {\C }^{k-1} \] for a unique \(m \in \Z \).
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