Classifying vector bundles

Proof. Fix a \(k\)-plane \(V\subseteq \C ^\infty \) and let \(U_V\) be the set of \(k\)-planes \(W\) for which the orthogonal projection \(W\to V\) is an isomorphism. Its intersection with every finite Grassmannian is open, so \(U_V\) is open in the colimit topology. Every \(W\in U_V\) is the graph of a unique linear map \(a_W\colon V\to V^\perp \), depending continuously on \(W\). The maps \[ U_V\times V\longrightarrow p_{\univ }^{-1}(U_V), \qquad (W,v)\longmapsto (W,v+a_W(v)) \] are therefore local trivializations of \(p_{\univ }\). □

Proof sketch of Theorem 9.2.4. We describe the main ingredients of Hatcher’s argument. The key observation is that, for a rank \(k\) vector bundle \(p\colon E \to X\), the following two types of data are equivalent:

(1)

A pair \((f,\phi )\) consisting of a continuous map \(f\colon X \to \Gr _k(\C ^\infty )\) and an isomorphism \(\phi \colon E \cong f^*(E_{\univ })\);

(2)

A continuous map \(g\colon E \to \C ^\infty \) whose restriction to each fiber is a \(\C \)-linear injection.

Indeed, an isomorphism \(E \cong f^*(E_{\univ })\) gives such a map by projecting \(f^*(E_{\univ }) \subseteq X \times \C ^\infty \) to \(\C ^\infty \). Conversely, a fiberwise linear injection \(g\) determines \(f(x) := g(E_x) \subseteq \C ^\infty \), and the evident map \(E \to E_{\univ }\) identifies \(E\) with \(f^*(E_{\univ })\).

Surjectivity is therefore reduced to constructing a map \(g\colon E\to \C ^\infty \) which is linear and injective on each fiber. Starting from a trivializing cover, one uses the countable-cover lemma for paracompact spaces to obtain a countable trivializing cover \((U_i)_{i\geq 1}\) together with a locally finite subordinate partition of unity \((\rho _i)_{i\geq 1}\). Let \(g_i\colon E|_{U_i}\to \C ^k\) be the coordinate map of a trivialization. The maps \[ v\longmapsto \rho _i(p(v))g_i(v), \] extended by zero outside \(U_i\), assemble into a map \(E\to (\C ^k)^\infty \cong \C ^\infty \). Local finiteness ensures that this map is continuous and takes values in the algebraic direct sum, while the partition-of-unity condition makes it injective on every fiber. This is the only point at which paracompactness is used in an essential way.

For injectivity, suppose two classifying maps \(f_0,f_1\colon X \to \Gr _k(\C ^\infty )\) pull back \(E_{\univ }\) to isomorphic bundles. The corresponding fiberwise injections \(g_0,g_1\colon E \to \C ^\infty \) may first be homotoped so that their images land in the odd and even coordinates, respectively. The straight-line homotopy \((1-t)g_0+tg_1\) then remains fiberwise injective for all \(t\), and hence determines a homotopy from \(f_0\) to \(f_1\).

For the countable-cover lemma, see Hatcher (2003), Lemma 1.21; the remaining continuity details are part of the proof of Hatcher (2003), Theorem 1.16. □

Proof. Local trivializations of \(P\) induce local trivializations of the associated bundle. Conversely, the frame bundle \(\mathrm {Fr}(E)\to B\) of a rank \(n\) vector bundle is a principal \(\GL _n(\C )\)-bundle. The natural map \(\mathrm {Fr}(E)\times _{\GL _n(\C )}\C ^n\to E\) is an isomorphism, and the map \[ P\longrightarrow \mathrm {Fr}(P\times _{\GL _n(\C )}\C ^n), \qquad p\longmapsto \big (v\mapsto [p,v]\big ) \] is an isomorphism of principal bundles. Thus the two constructions are inverse on isomorphism classes. □

Proof. By induction we may immediately reduce to the case \(n = 1\), i.e. we must show that the map \(\Phi _1\colon P \times G \to P \times _B P, \, (p,g) \mapsto (p,pg)\) is a homeomorphism. Observe that this map fits in a commutative diagram of the form

and so it suffices to find an open cover of \(P\) such that the given map restricts to a homeomorphism on the preimages. Since \(\pi \) is locally trivial, we may thus assume that it is of the form \(\pr \colon B \times G \to B\). But in this case, the map \(\Phi _1\) takes the form \(B \times G \times G \to B \times G \times G, \, (b,g,h) \mapsto (b,g,gh)\), which is indeed a homeomorphism with inverse \((b,g,h') \mapsto (b,g,g^{-1}h')\). □

Proof. It suffices to check the face and degeneracy maps. For the first face map, we have \begin {align*} \Phi _{n-1}(d_0^*(p,g_1,\dots ,g_n)) &=\Phi _{n-1}(pg_1,g_2,\dots ,g_n) \\ &=(pg_1,pg_1g_2,\dots ,pg_1g_2\dots g_n) =d_0^*\Phi _n(p,g_1,\dots ,g_n). \end {align*}

For a degeneracy map, inserting the identity element among the \(g_i\) inserts a repeated partial product among \(p,pg_1,\dots ,pg_1\dots g_n\), as required. The remaining face maps either multiply two adjacent \(g_i\) or omit the last one, and the same direct calculation proves compatibility. □

Proof. A principal bundle is a locally trivial fiber bundle and hence a Serre fibration by Theorem 2.3.13. Since \(\Pi _{\infty }\colon \Top \to \An \) preserves pullbacks along Serre fibrations by Proposition 2.3.19, there are isomorphisms \[ \Piinfty {P \times _B \dots \times _B P} \iso \Piinfty {P} \times _{\Piinfty {B}} \dots \times _{\Piinfty {B}} \Piinfty {P} \] for all \(n\), which assemble into an isomorphism of simplicial animae \(\Piinfty {\check {C}(\pi )} \iso \check {C}(\Piinfty {\pi })\). By Lemma 9.2.10, the simplicial anima \(\Piinfty {\check {C}(\pi )}\) is isomorphic to \([n] \mapsto \Piinfty {P \times G^n}\). By contractibility of \(P\), the projection maps \(\Piinfty {P \times G^n} \cong \Pi _{\infty }(P) \times \Piinfty {G^n} \to \Piinfty {G^n}\) are isomorphisms, which are compatible with the simplicial structure maps (see Chapterexercise 5.1). This proves part (1). For part (2), observe that \(\Pi _{\infty }(\pi )\) is surjective on path components, since \(\pi \) is surjective. It is therefore an effective epimorphism of animae, so \(\Pi _{\infty }(B)\) is the colimit of its Čech nerve. On the other hand, the colimit of the simplicial anima underlying the group object \(\Pi _{\infty }(G)\) is its classifying anima \(\bB G\). Part (1) therefore gives the claimed isomorphism. □

Proof. Let \(W \in \Gr _k(\C ^{\infty })\), and define \(U \subseteq \Gr _k(\C ^{\infty })\) to be the subspace of those \(k\)-planes \(V\) for which the orthogonal projection \(W \to V\) is an isomorphism. This is open: its intersection with every finite-dimensional stage \(\Gr _k(\C ^n)\) is open, and \(\Gr _k(\C ^{\infty })\) has the colimit topology. We claim that there exists a \(U(k)\)-equivariant homeomorphism \[ \phi \colon U \times U(k) \xrightarrow {\cong } \pi ^{-1}(U) \] over \(U\), which will suffice to finish the proof. To construct \(\phi \), fix a \(k\)-frame \((w_1, \dots , w_k)\) for \(W\). Given any \(k\)-plane \(V \in U\), the projection \(W \to V\) is an isomorphism, hence sends the \(k\)-frame \((w_1, \dots , w_k)\) to some basis for \(V\). Using Gram-Schmidt orthonormalization (which is a continuous operation), we may turn this into a \(k\)-frame \((v_1, \dots , v_k)\) for \(V\). All in all, we may then define \[ \phi (V,A) := (v_1, \dots , v_k) \circ A. \] Bijectivity of \(\phi \) holds because two orthonormal frames of a \(k\)-plane differ by a unique unitary \((k \times k)\)-matrix. We leave it to the reader to check that this is indeed a well-defined homeomorphism over \(U\). □

Proof. Consider the space \(\widetilde {V}_k(\C ^{\infty })\) of all linear injections \(\C ^k \hookrightarrow \C ^{\infty }\) which are not necessarily isometries. Equivalently, \(\widetilde {V}_k(\C ^{\infty })\) is the topological space of \(k\)-tuples \((v_1, \dots , v_k)\) of linearly independent vectors in \(\C ^{\infty }\) which are not necessarily orthogonal to each other. Observe that the inclusion \(V_k(\C ^{\infty }) \hookrightarrow \widetilde {V}_k(\C ^{\infty })\) is a homotopy equivalence: the Gram-Schmidt orthogonalization procedure provides a deformation retract \(\widetilde {V}_k(\C ^{\infty }) \to V_k(\C ^{\infty })\).

It will thus suffice to show that \(\widetilde {V}_k(\C ^{\infty })\) is contractible. To this end, consider the ‘shift map’ \(T^k\colon \C ^{\infty } \to \C ^{\infty }\) given on basis vectors by \(T^k(e_n) := e_{n+k}\). This induces a map \(T^k\colon \widetilde {V}_k(\C ^{\infty }) \to \widetilde {V}_k(\C ^{\infty })\). Note that this map is homotopic to the constant map on \((e_1, \dots , e_k)\) via the straight-line homotopy: \[ H(v,t) := (1-t)(e_1, \dots , e_k) + t T^k(v). \] This is well-defined since the first \(k\) components of the vectors \(T^k(v_i)\) are all zero. Similarly, the map \(T^k\) is homotopic to the identity on \(\widetilde {V}_k(\C ^{\infty })\), again via the straight-line homotopy: \[ H(v,t) := (1-t)v + t T^k(v). \] To see that this is well-defined, assume for contradiction that for some time \(t\) the vectors are no longer linearly independent. Then there are scalars \(\lambda _1, \dots , \lambda _k\), not all zero, such that \(\sum _{i=1}^k \lambda _i ((1-t)v_i + t T^k(v_i)) = 0\). But then the nonzero vector \(w := \sum _{i=1}^k\lambda _i v_i\) satisfies \((1-t) w + t T^k(w) = 0\). For \(t<1\), this is impossible by considering the first nonzero coordinate of \(w\); for \(t=1\), it is impossible because \(T^k\) is injective. This contradicts linear independence of \(v\).

It follows that the identity on \(\widetilde {V}_k(\C ^{\infty })\) is homotopic to a constant map, and hence this space is contractible. □

Proof. In light of Lemma 9.2.12 and Lemma 9.2.14, this is an instance of Proposition 9.2.11. □

Proof. Both sides of the first bijection decompose according to locally constant functions \(r\colon X\to \N \). For such a function, let \(X_k:=r^{-1}(k)\). These subspaces are clopen, and hence paracompact Hausdorff, and they form a topological disjoint-union decomposition \[ X=\bigsqcup _{k\geq 0}X_k. \] A continuous map \(X\to \bigsqcup _{k\geq 0}\Gr _k(\C ^\infty )\) with component function \(r\) is precisely a family of continuous maps \(X_k\to \Gr _k(\C ^\infty )\). Similarly, a vector bundle with rank function \(r\) is precisely a family of rank \(k\) vector bundles over the \(X_k\). The first bijection therefore follows on each clopen stratum from Theorem 9.2.4.

The Schubert cell structures above also make \(\bigsqcup _{k\geq 0}\Gr _k(\C ^\infty )\) a cell complex. If \(X\) is a cell complex as well, then Corollary 2.3.7, Corollary 9.2.15 identifies the source of the first bijection with the source of the second. □

Proof. Assume for contradiction that \(K\) is not contained in any \(Y_n\), and choose \(x_n \in K \setminus Y_n\). Set \(A:=\{x_n\mid n\geq 0\}\). This set is infinite: if it were finite, all its points would lie in some common stage \(Y_N\), contradicting \(x_n\notin Y_n\) for \(n\geq N\). For every \(m\), the intersection \(A\cap Y_m\) is contained in the finite set \(\{x_0,\dots ,x_{m-1}\}\), hence is closed in the \(T_1\)-space \(Y_m\). The same holds for every subset of \(A\). By the definition of the colimit topology, every subset of \(A\) is therefore closed in \(Y\). Thus \(A\) is an infinite closed discrete subspace of \(K\), contradicting compactness of \(K\).

The last claim follows by applying the first to the compact subset \(f(X) \subseteq Y\). □

Proof. We use the Whitehead theorem for animae. The compactness lemma, applied to the point and to the interval, shows that the canonical morphism induces a bijection on path components. Fix a basepoint represented by some \(y\in Y_N\). For every \(q\geq 1\), compactness of \(S^q\) and \(S^q\times [0,1]\) shows that maps \(S^q\to Y\) and homotopies between them factor through finite stages. Consequently, \[ \colim _{n\geq N}\pi _q(Y_n,y) \xrightarrow {\cong } \pi _q(Y,y). \] By Remark 2.4.19, these are the homotopy groups of the corresponding underlying animae. Homotopy groups of pointed animae commute with filtered colimits, so the left-hand side is also the \(q\)-th homotopy group of \(\colim _n\Pi _{\infty }(Y_n)\) at \(y\). The claim now follows from Proposition 2.4.22. □

Proof. The principal bundles \(V_n(\C ^\infty ) \to \Gr _n(\C ^\infty )\) identify \(\Pi _{\infty }\Gr _n(\C ^\infty )\) with \(\bB U(n)\) by Corollary 9.2.15. These identifications are compatible with the stabilization maps \(\Gr _n(\C ^\infty ) \to \Gr _{n+1}(\C ^\infty )\) and the block-sum inclusions \(U(n) \to U(n+1)\). Since the Grassmannian stabilization maps are closed embeddings between Hausdorff spaces, Proposition 9.2.19 gives \[ \Pi _{\infty }(BU) \simeq \colim _n \bB U(n). \] The classifying-anima equivalence \(\bB \colon \Grp (\An )\iso \An _{*,\geq 1}\) preserves colimits. Since connected pointed animae are closed under filtered colimits, the filtered colimit of the \(\bB U(n)\) agrees whether computed in \(\An \), \(\An _*\), or \(\An _{*,\geq 1}\). Moreover, filtered colimits in \(\Grp (\An )\) are computed on underlying animae, since finite limits commute with filtered colimits in \(\An \). Applying Proposition 9.2.19 also to the closed embeddings \(U(n)\hookrightarrow U(n+1)\) therefore gives \[ \colim _n\bB U(n) \simeq \bB \!\left (\colim _n\Pi _{\infty }U(n)\right ) \simeq \bB \Pi _{\infty }(U), \] which is the asserted classifying anima \(\bB U\). □