2.1 Orthogonal groups, Grassmannians and Stiefel manifolds

2.1.1 Main definitions

We start by recalling the definition of the Frobenius norm. Let $l,m \in \mathbb {N}_{\geq 1}$ , and let $\mathbb {R}^{l \times m}$ denote the set of $l \times m$ matrices with real coefficients. Let $A \in \mathbb {R}^{l \times m}$ . The Frobenius norm of A is defined by

$$\begin{align*}\| A\| = \sqrt{\operatorname{\mathrm{tr}}(A^t A)} = \sqrt{\sum_{ 1 \leq i \leq l, 1 \leq j \leq m} A_{ij}^2}, \end{align*}$$

where $A^t \in \mathbb {R}^{m \times l}$ denotes the transpose of A. The Frobenius norm, as any norm, induces a distance on $\mathbb {R}^{l \times m}$ defined by $\mathsf {d}_{\mathsf {Fr}}\left ( A, B \right ) = \|A - B\|$ . We refer to this distance as the Frobenius distance.

We now introduce the spaces of matrices we are most interested in. Let $n\geq d \geq 1 \in \mathbb {N}$ .

The Grassmannian $\mathsf {Gr}(d,n)$ has as elements the $n \times n$ real matrices A that satisfy $A = A^t = A^2$ and have rank equal to d and is thus a subset of $\mathbb {R}^{n \times n}$ . We metrize and topologize $\mathsf {Gr}(d,n)$ using the Frobenius distance. Note that the elements of $\mathsf {Gr}(d,n)$ are canonically identified with the orthogonal projection operators $\mathbb {R}^n \to \mathbb {R}^n$ of rank d. Since these orthogonal projections are completely determined by the subspace of $\mathbb {R}^n$ which they span, it follows that the elements of $\mathsf {Gr}(d,n)$ correspond precisely to the d-dimensional subspaces of $\mathbb {R}^n$ .

The (compact) Stiefel manifold $\mathsf {V}(d,n)$ has as elements the set of $n \times d$ real matrices with orthonormal columns. We metrize $\mathsf {V}(d,n)$ with the Frobenius distance. Note that the elements of $\mathsf {V}(d,n)$ can be identified with the d-dimensional subspaces of $\mathbb {R}^n$ equipped with an ordered orthonormal basis. The elements of a Stiefel manifold are sometimes referred to as frames.

When $d = n$ , the Stiefel manifold $\mathsf {V}(d,n)$ coincides with the orthogonal group $O(d)$ , which consists of real $d \times d$ matrices $\Omega $ such that $\Omega \Omega ^t = \mathsf {id}$ . We metrize $O(d)$ using the Frobenius distance. With this definition, $O(d)$ is a topological group, as matrix multiplication and inversion are continuous.

We remark here that, although we shall encounter other metrics for $O(d)$ and $\mathsf {Gr}(d,n)$ , our main results are stated using the Frobenius distance. The relevant results about other metrics on the orthogonal group can be found in Appendix A.4, and the ones about other metrics on the Grassmannian are in Appendix A.4.

2.1.2 Infinite-dimensional Grassmannians and Stiefel manifolds

Let $n\geq d \geq 1 \in \mathbb {N}$ . There is an inclusion $\mathsf {Gr}(d,n) \subseteq \mathsf {Gr}(d,n+1)$ given by adding a row and a column of zeros at the bottom and right, respectively. This inclusion is norm-preserving and thus metric-preserving. We define the infinite-dimensional Grassmannian as $\mathsf {Gr}(d) = \bigcup _{n \in \mathbb {N}} \mathsf {Gr}(d,n)$ . We topologize $\mathsf {Gr}(d)$ using the direct limit topology; recall that the direct limit topology on a union $X = \bigcup _{n \in \mathbb {N}} X_n$ of topological spaces $\{X_n\}_{n \in \mathbb {N}}$ , where $X_n$ is a subspace of $X_{n+1}$ for all $n \in \mathbb {N}$ , is the topology where $A \subseteq X$ is open if and only if $A \cap X_n$ is open in $X_n$ for all $n \in \mathbb {N}$ . Note that this topology is finer (i.e., has more opens) than the topology induced by the metric inherited by $\mathsf {Gr}(d)$ by virtue of it being an increasing union of metric spaces.

Similarly, we have an inclusion $\mathsf {V}(d,n) \subseteq \mathsf {V}(d,n+1)$ given by taking the matrix representation of a d-frame in $\mathbb {R}^n$ and adding a row of zeros at the bottom of the matrix. Again, these inclusions are metric preserving, and we define $\mathsf {V}(d) = \bigcup _{n \in \mathbb {N}} \mathsf {V}(d,n)$ , with the direct limit topology induced by the inclusions $\mathsf {V}(d,n) \to \mathsf {V}(d)$ .

There is a principal $O(d)$ -bundle $\mathsf {Proj} : \mathsf {V}(d,n) \to \mathsf {Gr}(d,n)$ , defined by mapping a d-frame M to the matrix $MM^t$ (see Section 2.2.2 for the notion of principal bundle). Note that the maps $\mathsf {Proj} : \mathsf {V}(d,n) \to \mathsf {Gr}(d,n)$ for each $n \geq 1$ assemble into a map

$$\begin{align*}\mathsf{Proj} : \mathsf{V}(d) \to \mathsf{Gr}(d). \end{align*}$$

It is clear that $\mathsf {Proj}$ is continuous, as it restricts to a continuous map $\mathsf {Proj} : \mathsf {V}(d,n) \to \mathsf {Gr}(d,n)$ for each $n \geq d$ .

2.2 Principal bundles and vector bundles

We recall the main notions and results that are relevant to this article. For a thorough exposition, we refer the reader to, for example, [38, Appendices A and B], [Reference Steenrod61], [Reference Husemöller34] and [Reference Husemöller, Joachim, Jurčo, Schottenloher, Echterhoff, Fredenhagen and Krötz33]. The classic book [Reference Milnor and Stasheff47] contains all of the standard results on vector bundles we need; see also [Reference Hatcher27] for a modern exposition.

2.2.3 Čech cocycles

Let B be a topological space, G a topological group and $\mathcal {U}$ a cover of B. A Čech $1$ -cocycle subordinate to $\mathcal {U}$ with coefficients in G consists of a family of continuous maps $\{\rho _{ij} : U_j \cap U_i \to G\}_{(ij) \in N(\mathcal {U})}$ indexed by the ordered $1$ -simplices of $N(\mathcal {U})$ , which satisfies the cocycle condition, meaning that for every $(ijk) \in N(\mathcal {U})$ and $y \in U_k \cap U_j \cap U_i$ , we have

$$\begin{align*}\rho_{ij}(y) \rho_{jk}(y) = \rho_{ik}(y) \in G. \end{align*}$$

We remark that we are using a convention for defining $\rho _{ij}$ so that $\rho _{ij}$ and $\rho _{jk}$ can be composed from left to right. The opposite convention is also common. The set of $1$ -cocycles subordinate to $\mathcal {U}$ with coefficients in G is denoted by $Z^1(\mathcal {U};G)$ .

The following construction associates a cocycle to any principal G-bundle and motivates the notion of cocycle. Given a principal G-bundle over B, there exists, by definition, a cover $\mathcal {U} = \{U_i\}_{i \in I}$ and a family of G-equivariant local trivializations $\{\rho _i : U_i \times G \to p^{-1}(U_i)\}_{i \in I}$ . Note that, whenever $i,j \in I$ are such that $U_j \cap U_i \neq \emptyset $ , the map $\rho _j^{-1} \circ \rho _i : (U_j \cap U_i) \times G \to (U_j \cap U_i) \times G$ is G-equivariant. It follows that $\rho _j^{-1} \circ \rho _i$ induces a continuous map $\rho _{ij} : U_j \cap U_i \to G$ satisfying $(\rho _j^{-1}\circ \rho _i)(y,g) = (y, g \cdot \rho _{ij}(y))$ for all $(y,g) \in (U_j \cap U_i) \times G$ , and that the family $\{\rho _{ij} : U_j \cap U_i \to G\}_{(ij) \in N(\mathcal {U})}$ satisfies the cocycle condition.

2.2.4 Čech cohomology

Let $\mathcal {U} = \{U_i\}_{i \in I}$ be a cover of B. A Čech $\boldsymbol {0}$ -cochain subordinate to $\mathcal {U}$ with values in G consists of a family of continuous maps $\Theta = \{\Theta _i : U_i \to G\}_{i \in I}$ . Let $C^0(\mathcal {U};G)$ denote the set of $0$ -cochains subordinate to $\mathcal {U}$ with values in G. There is an action $C^0(\mathcal {U};G) \curvearrowright Z^1(\mathcal {U};G)$ with $\Theta = \{\Theta _i : U_i \to G\}_{i \in I}$ acting on a cocycle $\rho = \{\rho _{ij} : U_j \cap U_i \to G\}_{(ij) \in N(\mathcal {U})}$ by $\Theta \cdot \rho = \{\Theta _i \rho _{ij} \Theta _j^{-1} : U_j \cap U_i \to G\}_{(ij) \in N(\mathcal {U})}$ . The quotient of $Z^1(\mathcal {U};G)$ by the action of $C^0(\mathcal {U};G)$ is denoted by $\check {\mathsf {H}}^{1}_{}\left ( \mathcal {U}; G \right )$ .

Let $\mathcal {U} = \{U_i\}_{i \in I}$ and $\mathcal {V} = \{V_j\}_{j \in J}$ be two covers of a common topological space B. A refinement $\nu : \mathcal {U} \to \mathcal {V}$ consists of a function $\nu : I \to J$ such that for every $i \in I$ we have $U_i \subseteq V_{\nu (i)}$ . Given a refinement $\nu : \mathcal {U} \to \mathcal {V}$ and $\rho \in Z^1(\mathcal {V};G)$ , define a cocycle $\nu (\rho ) \in Z^1(\mathcal {U};G)$ by $\nu (\rho )_{ij} = \rho _{\nu (i)\nu (j)}$ . It is not hard to check that any two refinements $\nu , \nu ' : \mathcal {U} \to \mathcal {V}$ induce the same map $\check {\mathsf {H}}^{1}_{}\left ( \mathcal {V}; G \right ) \to \check {\mathsf {H}}^{1}_{}\left ( \mathcal {U}; G \right )$ . One then defines the Čech cohomology of B with coefficients in G as

$$\begin{align*}\check{\mathsf{H}}^{1}_{}\left( B; G \right) = \operatorname*{\mathrm{colim}}_{\mathcal{U}} \check{\mathsf{H}}^{1}_{}\left( \mathcal{U}; G \right), \end{align*}$$

where the colimit is indexed by the poset whose objects are covers of B and where $\mathcal {V} \preceq \mathcal {U}$ if there exists a refinement $\mathcal {U} \to \mathcal {V}$ .

As we saw previously, any principal G-bundle over B can be trivialized over some cover of B and induces a cocycle over that cover. It is well known, and easy to see, that this construction induces a bijection

$$\begin{align*}\mathsf{Prin}_G(B) \to \check{\mathsf{H}}^{1}_{}\left( B; G \right). \end{align*}$$

For a description of principal G-bundles from this point of view, see [Reference Lawson and Michelsohn38, Appendix A].

2.2.5 Vector bundles

Let $d \in \mathbb {N}_{\geq 1}$ . A vector bundle of rank d over B consists of a fiber bundle $p : E \to B$ with fiber $\mathbb {R}^d$ , where each fiber comes with the structure of a real vector space of dimension d and such that p admits a local trivialization that is linear on each fiber.

It follows that $\rho _j^{-1} \circ \rho _i$ induces a well-defined continuous map $\rho _{ij} : U_j \cap U_i \to GL(d)$ satisfying $(\rho _j^{-1}\circ \rho _i)(y,g) = (y, g \cdot \rho _{ij}(y))$ for all $(y,g) \in (U_j \cap U_i) \times \mathbb {R}^d$ . So every vector bundle of rank d over B that trivializes over a cover $\mathcal {U}$ gives a cocycle in $Z^1(\mathcal {U};GL(d))$ .

An isomorphism between vector bundles $p : E \to B$ and $p' : E' \to B$ consists of a fiberwise map $m : E \to E'$ that is a linear isomorphism on each fiber. The family of isomorphism classes of rank-d vector bundles over B is denoted by $\mathsf {Vect}_d(B)$ .

A partition of unity argument shows that, if B is paracompact (in the sense of [Reference Milnor and Stasheff47, Section 5.8]), the trivialization of a vector bundle on B can be taken so that the associated cocycle takes values in the orthogonal group. When B is paracompact, this gives a bijection

$$\begin{align*}\mathsf{Vect}_d(B) \to \check{\mathsf{H}}^{1}_{}\left( B; O(d) \right). \end{align*}$$

In particular, $\mathsf {Vect}_d(B) \cong \check {\mathsf {H}}^{1}_{}\left ( B; O(d) \right ) \cong \mathsf {Prin}_{O(d)}(B)$ .

2.2.6 Classifying maps

For details about the claims in this section, see [Reference Milnor45, Theorem 3.1] for the existence of classifying spaces of topological groups and [Reference Lawson and Michelsohn38, Appendix B] for an account of classifying spaces of Lie groups.

For every topological group G, there exists a classifying space $\mathcal {B} G$ , which consists of a topological space with the property that, for every paracompact topological space B, there is a natural bijection

$$\begin{align*}\mathsf{Prin}_G(B)\cong \left\lbrack {B}, {\mathcal{B} G} \right\rbrack. \end{align*}$$

The classifying space of the orthogonal group $O(d)$ is the Grassmannian $\mathsf {Gr}(d)$ , and thus

$$\begin{align*}\mathsf{Vect}_d(B) \cong \check{\mathsf{H}}^{1}_{}\left( B; O(d) \right) \cong \mathsf{Prin}_{O(d)}(B) \cong \left\lbrack {B}, {\mathsf{Gr}(d)} \right\rbrack. \end{align*}$$

The map $\left \lbrack {B}, {\mathsf {Gr}(d)} \right \rbrack \to \mathsf {Prin}_{O(d)}(B)$ is constructed as follows. Given $[f] \in \left \lbrack {B}, {\mathsf {Gr}(d)} \right \rbrack $ , let $f : B \to \mathsf {Gr}(d)$ be a representative. Then, the pullback of $\mathsf {Proj} : \mathsf {V}(d) \to \mathsf {Gr}(d)$ along f is a principal $O(d)$ -bundle over B, whose isomorphism type is independent of the choice of representative for $[f]$ .

3.1 Approximate cocycles

The notion of approximate cocycle makes sense for any metric group, which we define next. This extra generality makes some arguments clearer and will be of use in Section 6.1.

The main example of metric group to keep in mind is that of the orthogonal group $O(d)$ endowed with the Frobenius distance. Other relevant examples include all connected Lie groups endowed with the geodesic distance induced by a bi-invariant Riemannian metric; recall that all compact Lie groups, such as the orthogonal groups, the unitary groups and the compact symplectic groups, admit a bi-invariant Riemannian metric (see, e.g., [Reference Milnor46, Corollary 1.4]). It is also relevant to note that the Frobenius distance on $O(d)$ does not arise as the geodesic distance induced by a Riemannian metric; Section A.4 deals with some of the relationships between the Frobenius distance and the (usual) geodesic distance on $O(d)$ .

Let G be a metric group and let us denote its distance by $d_G : G \times G \to \mathbb {R}$ . Let B be a topological space and let $\mathcal {U} = \{U_i\}_{i \in I}$ be a cover of B.

We denote the set of all $1$ -cochains on B subordinate to $\mathcal {U}$ with values in G by $C^1(\mathcal {U};G)$ . If there is no risk of confusion, we may refer to an element of $C^1(\mathcal {U};G)$ simply as a cochain subordinate to $\mathcal {U}$ .

We denote the set of $\varepsilon $ -approximate cocycles by $\mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; G \right ) \subseteq C^1(\mathcal {U};G)$ , and the set of exact cocycles by either $\mathsf {Z}^{1}_{0}\left ( \mathcal {U}; G \right )$ or simply $\mathsf {Z}^{1}_{}\left ( \mathcal {U}; G \right )$ . Of course, when $\varepsilon = \infty $ , $\varepsilon $ -approximate cocycles are merely cochains, so to keep notation uniform, we denote $C^1(\mathcal {U};G)$ by $\mathsf {Z}^{1}_{\infty }\left ( \mathcal {U}; G \right )$ . We endow the set $\mathsf {Z}^{1}_{\infty }\left ( \mathcal {U}; G \right )$ with the metric given by

$$\begin{align*}\mathsf{d}_{\mathsf{Z}}\left( \Omega, \Lambda \right) := \sup_{(ij) \in N(\mathcal{U})} \sup_{y \in U_j \cap U_i} d_G\left(\Omega_{ij}(y), \Lambda_{ij}(y) \right), \end{align*}$$

for $\Omega ,\Lambda \in \mathsf {Z}^{1}_{\infty }\left ( \mathcal {U}; G \right )$ . This induces a metric on all spaces of approximate and exact cocycles. In particular, for $\varepsilon \leq \varepsilon '$ , we have metric embeddings

$$\begin{align*}\mathsf{Z}^{1}_{}\left( \mathcal{U}; G \right) \subseteq \mathsf{Z}^{1}_{\varepsilon}\left( \mathcal{U}; G \right) \subseteq \mathsf{Z}^{1}_{\varepsilon'}\left( \mathcal{U}; G \right) \subseteq \mathsf{Z}^{1}_{\infty}\left( \mathcal{U}; G \right). \end{align*}$$

We denote the set of all $0$ -cochains by $C^0(\mathcal {U};G)$ . The set $C^0(\mathcal {U};G)$ forms a group, by pointwise multiplication. There is an action $C^0(\mathcal {U};G) \curvearrowright \mathsf {Z}^{1}_{\infty }\left ( \mathcal {U}; G \right )$ with $\Theta $ acting on $\Omega $ by

$$\begin{align*}(\Theta \cdot \Omega)_{ij}(y) = \Theta_i(y)\, \Omega_{ij}(y)\, \Theta_j^{-1}(y)\end{align*}$$

for every $y \in U_j \cap U_i$ . Since G acts on itself by isometries, the above action restricts to an action on $\mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; G \right )$ for every $\varepsilon \geq 0$ .

Since the action $C^0(\mathcal {U};G) \curvearrowright \mathsf {Z}^{1}_{\infty }\left ( \mathcal {U}; G \right )$ is by isometries, the set $\check {\mathsf {H}}^{1}_{\varepsilon }\left ( \mathcal {U}; G \right )$ inherits a metric $\mathsf {d}_{\check {\mathsf {H}}}$ from $\mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; G \right )$ , given as follows. For $\Omega ,\Lambda \in \check {\mathsf {H}}^{1}_{\varepsilon }\left ( \mathcal {U}; G \right )$ , we have

$$\begin{align*}\mathsf{d}_{\check{\mathsf{H}}}\left( \Omega, \Lambda \right) := \inf_{\overline{\Omega},\overline{\Lambda} \in \mathsf{Z}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right)} \mathsf{d}_{\mathsf{Z}}\left( \overline{\Omega}, \overline{\Lambda} \right), \end{align*}$$

where $\overline {\Omega }$ and $\overline {\Lambda }$ range over all representatives of $\Omega $ and $\Lambda $ , respectively.

3.2 Approximate local trivializations

We start with some considerations about the notion of local trivialization of a vector bundle (Section 2.2.5), which motivate our notion of approximate local trivialization. Any rank-d Euclidean vector bundle $p : E \to B$ admits an isometric local trivialization, that is, a local trivialization over an open cover $\mathcal {U} = \{U_i\}_{i \in I}$ of B given by a family of homeomorphisms $\{U_i \times \mathbb {R}^d \to p^{-1}(U_i) \}_{i \in I}$ with the property that, given $y \in U_i$ , we obtain an orthonormal basis of the vector space $p^{-1}(y)$ by evaluating the map $U_i \times \mathbb {R}^d \to p^{-1}(U_i)$ on $(y,e_j)$ for $1 \leq j \leq d$ , where $e_j$ is the jth canonical basis vector of $\mathbb {R}^d$ . Recall that any vector bundle over a paracompact base B is classified by some continuous map $B \to \mathsf {Gr}(d)$ (Section 2.2.6). This implies, in particular, that, up to isomorphism of vector bundles, any vector bundle $p : E \to B$ over a paracompact base is a Euclidean vector bundle whose fibers $p^{-1}(y)$ that are not just abstract vector spaces, but d-dimensional subspaces of $\mathbb {R}^\infty $ .

With the above in mind, any isometric local trivialization of a Euclidean vector bundle over a paracompact base B gives us maps $\{\Phi _i : U_i \to \mathsf {V}(d)\}$ , where, as in Section 2.1.1, the space $\mathsf {V}(d)$ stands for the Stiefel manifold of d-frames in $\mathbb {R}^\infty $ . One can check that a family of maps $\{\Phi _i : U_i \to \mathsf {V}(d)\}_{i \in I}$ comes from a vector bundle over B precisely when, for every intersection $U_j \cap U_i \neq \emptyset $ and every $y \in U_j \cap U_i$ , we have that $\Phi _i(y)$ and $\Phi _j(y)$ span the same subspace of $\mathbb {R}^\infty $ and thus, equivalently, when there exist continuous maps $\{\Omega _{ij} : U_j \cap U_i \to O(d)\}_{(ij) \in N(\mathcal {U})}$ such that $\Phi _i(y) \Omega _{ij}(y) = \Phi _j(y)$ for all $y \in U_j \cap U_i$ . Our notion of approximate local trivialization is based on this last equivalent characterization of local trivializations and, specifically, on relaxing this last equality.

Let $\mathcal {U} = \{U_i\}_{i \in I}$ be a cover of a topological space B.

We say that the family $\Omega = \{\Omega _{ij}\}_{(ij) \in N(\mathcal {U})}$ in Definition 3.8 is a witness of the fact that $\Phi $ is an $\varepsilon $ -approximate (or exact) local trivialization. We remark that this witness is not part of the data of an $\varepsilon $ -approximate local trivialization and that we merely require that a witness exists.

An approximate local trivialization consists of an $\varepsilon $ -approximate local trivialization for some $\varepsilon \in [0,\infty ]$ . We denote the set of $\varepsilon $ -approximate local trivializations subordinate to $\mathcal {U}$ by $\mathsf {T}_{\varepsilon }\left ( \mathcal {U}; d \right )$ . We define a metric on $\mathsf {T}_{\varepsilon }\left ( \mathcal {U}; d \right )$ by

$$\begin{align*}\mathsf{d}_{\mathsf{T}}\left( \Phi, \Psi \right) := \sup_{i \in I} \sup_{y \in U_i} \|\Phi_i(y) - \Psi_i(y)\|. \end{align*}$$

An approximate local trivialization $\{\Phi _i\}_{i \in I}$ is nondegenerate if, for every $(ij) \in N(\mathcal {U})$ and every $y \in U_j \cap U_i$ , the $d\times d$ matrix $\Phi _i(y)^t \Phi _j(y)$ has full rank.

3.3 Approximate classifying maps

Recall from Section 2.1.3 the definition of the thickened Grassmannians, and recall, in particular, that the topology of $\mathsf {Gr}(d)^\varepsilon $ is the direct limit topology and not the one induced by the Frobenius metric. Let B be a topological space.

The set of $\varepsilon $ -approximate classifying maps is denoted by $\mathsf {Maps}\left ( {B}, {\mathsf {Gr}(d)^\varepsilon } \right )$ . We define a metric on this set by

$$\begin{align*}\mathsf{d}_{\mathsf{C}}\left( f, g \right) = \sup_{y \in B} \| f(y)-g(y)\|. \end{align*}$$

We are also interested in the set of classifying maps up to homotopy, which we denote by $\left \lbrack {B}, {\mathsf {Gr}(d)^\varepsilon } \right \rbrack $ . Although we will not make use of this fact, we mention that one can interpret this as a persistent set $\left \lbrack {B}, {\mathsf {Gr}(d)^{-}} \right \rbrack : ([0,\infty ], \leq ) \to \mathbf {Set}$ , in the sense of [Reference Curry15, Definition 2.2]. The following result is easily proven using a linear homotopy.

4.1 Cocycles and local trivializations

In this section, we relate approximate cocycles and approximate local trivializations. We give constructions (Construction 4.2 and Construction 4.8) to go back and forth between the notions, and we show that, in a sense, these constructions are approximate inverses of each other (Remark 4.11). The construction to go from approximate local trivializations to approximate cocycles is in general not canonical; we conclude the section by showing that, when $\varepsilon \leq 1$ , the construction can be made canonical.

To motivate the assumptions made in the following construction, recall that a set I is countable if there exists an injection $\iota : I \to \mathbb {N}_{\geq 1}$ . Recall also that any vector bundle on a paracompact topological space can be trivialized on a countable open cover [Reference Milnor and Stasheff47, Lemma 5.9], and that every open cover of a paracompact topological space admits a subordinate partition of unity.

We start with a simplification. Given $\mathcal {V} = \{V_i\}_{i \in I}$ a countable open cover of a topological space B and an injection $\iota : I \to \mathbb {N}_{\geq 1}$ , consider a new open cover $\mathcal {U} = \iota _*(\mathcal {V})$ of B indexed by $\mathbb {N}_{\geq 1}$ with $U_n = V_i$ if $\iota (i) = n$ or $U_n = \emptyset $ if there is no $i \in I$ such that $\iota (i) = n$ .

We give the main constructions of this section for covers indexed by $\mathbb {N}_{\geq 1}$ , and we will later generalize them to arbitrary countable covers, as this simplifies exposition.

Construction 4.2. Let B be a paracompact topological space and let $\mathcal {V} = \{V_i\}_{i \in \mathbb {N}_{\geq 1}}$ be a cover of B. Let $\varphi $ be a partition of unity subordinate to $\mathcal {V}$ . Given a cochain $\Omega $ subordinate to $\mathcal {V}$ define, for each $i \in \mathbb {N}_{\geq 1}$ , a map $\Phi _i : V_i \to \mathsf {V}(d)$ , where the rows of $\Phi _i(y)$ from $d\times j$ to $d \times (j+1)-1$ are given by

$$\begin{align*}\sqrt{\varphi_j(y)} \Omega_{ij}(y)^t. \\[-36pt] \end{align*}$$

$\vartriangleleft $

Note that the maps $\Phi _i$ of Construction 4.2 are continuous and are well defined since, if $y \not \in V_j$ , then $\varphi _j(y) = 0$ .

Proof. We give the proof for $\varepsilon \in (0,\infty ]$ , the case $\varepsilon = 0$ being similar. Let $(ij) \in N(\mathcal {V})$ . We claim that the original $\varepsilon $ -approximate cocycle $\Omega $ is a witness that the family $\Phi $ is an $\varepsilon $ -approximate local trivialization. To prove this, we must show that, for all $y \in V_j \cap V_i$ , the Frobenius distance between $\Phi _i(y) \Omega _{ij}(y)$ and $\Phi _j(y)$ is less than $\varepsilon $ . Carrying out the product $\Phi _i(y) \Omega _{ij}(y)$ , we get an element of $\mathsf {V}(d)$ with rows from $d\times k$ to $d \times (k+1)-1$ given by

$$\begin{align*}\sqrt{\varphi_k(y)} \Omega_{ik}(y)^t \Omega_{ij}(y).\end{align*}$$

So $\|\Phi _i(y) \Omega _{ij}(y) - \Phi _j(y)\| = \left (\sum _{k \geq 1} \| \sqrt {\varphi _k(y)} \left (\Omega _{ik}(y)^t \Omega _{ij}(y) - \Omega _{kj}(y)\right )\| ^2\right )^{1/2} < \varepsilon $ , as required.

We have thus constructed a map

$$\begin{align*}\mathsf{triv}^\varphi : \mathsf{Z}^{1}_{\varepsilon}\left( \mathcal{V}; O(d) \right) \to \mathsf{T}_{\varepsilon}\left( \mathcal{V}; d \right), \end{align*}$$

for any cover $\mathcal {V}$ of a paracompact topological space B that is indexed by $\mathbb {N}_{\geq 1}$ . It is important to note that this map depends on the choice of partition of unity $\varphi $ . Nevertheless, using two different partitions of unity gives homotopic local trivializations, in the following sense.

Next, we show that the construction assigning an approximate local trivialization to an approximate cocycle is stable.

Lemma 4.5. Let $\mathcal {V} = \{V_i\}_{i \in \mathbb {N}_{\geq 1}}$ be a cover of a paracompact topological space B, and let $\varphi $ be a partition of unity subordinate to $\mathcal {V}$ . For $\Omega $ and $\Lambda\ \varepsilon $ -approximate cocycles subordinate to $\mathcal {V}$ ,

$$\begin{align*}\mathsf{d}_{\mathsf{T}}\left( \mathsf{triv}^\varphi(\Omega), \mathsf{triv}^\varphi(\Lambda) \right) \leq \mathsf{d}_{\mathsf{Z}}\left( \Omega, \Lambda \right). \end{align*}$$

Proof. Let $\mathsf {d}_{\mathsf {Z}}\left ( \Omega , \Lambda \right ) = \delta $ , $\Phi := \mathsf {triv}^\varphi (\Omega )$ , and $\Psi := \mathsf {triv}^\varphi (\Lambda )$ . For $y \in V_i$ , we have

$$\begin{align*}\| \Phi_i(y) - \Psi_i(y)\| ^2 = \sum_{k \in I} \varphi_k(y) \| \Omega_{ik}(y) - \Lambda_{ik}(y)\| ^2 \leq \delta^2 \sum_{k \in I} \varphi_k(y) = \delta^2, \end{align*}$$

which proves the claim.

Let $\mathcal {U} = \{U_i\}_{i \in I}$ be a countable open cover of a paracompact topological space B, and let $\iota : I \to \mathbb {N}_{\geq 1}$ be an injection. Using Remark 4.1, we can generalize Construction 4.2, Lemma 4.3, Lemma 4.4 and Lemma 4.5 to $\mathcal {U}$ .

In particular, we have a map $\mathsf {triv}^{\varphi ,\iota } : \mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right ) \to \mathsf {T}_{\varepsilon }\left ( \mathcal {U}; d \right )$ that now also depends on the choice of injection $\iota $ . We now prove that, up to homotopy, $\mathsf {triv}$ is independent of the choice of $\iota $ . In order to show this, we prove a more general lemma that will be of use later.

The proof of the lemma is standard, but we give it here for completeness.

Proof. Assume that $\iota $ and $\iota '$ have disjoint images. Then, there is a homotopy between $\chi ^\iota $ and $\chi ^{\iota '}$ given by $\sqrt {\alpha } \chi ^{\iota } + \sqrt {1-\alpha } \chi ^{\iota '}$ for $\alpha \in [0,1]$ . Let $2\iota : \mathbb {N}_{\geq 1} \to \mathbb {N}_{\geq 1}$ be given by $2\iota (k) = 2 \times \iota (k)$ and define $2\iota ' + 1$ in an analogous way. Since $2\iota $ and $2\iota '+1$ have disjoint images, our previous reasoning reduces the problem to showing that $\chi ^{\iota }$ and $\chi ^{2\iota }$ are homotopic and that $\chi ^{\iota '}$ and $\chi ^{2\iota ' - 1}$ are homotopic. Since the two proofs are entirely analogous, we give the details only for the case of $\iota $ . Moreover, since $2\iota : \mathbb {N}_{\geq 1} \to \mathbb {N}_{\geq 1}$ is the composite of $\iota $ with multiplication by $2$ , it is enough to give the proof for $\iota = \mathsf {id}$ , which we now do.

We must show that the identity $\mathsf {V}(d) \to \mathsf {V}(d)$ is homotopic to $\chi ^2 : \mathsf {V}(d) \to \mathsf {V}(d)$ . Informally, the proof works by moving each row of $\Phi $ at a time. To simplify exposition, in the rest of this proof, the notation $\Phi _m$ will be used to refer to the mth row of a frame $\Phi \in \mathsf {V}(d)$ . Consider the family of functions $f^\alpha : \mathsf {V}(d) \to \mathsf {V}(d)$ indexed by $\alpha \in [0,1]$ defined as follows. For $\alpha = 0$ , let $f^\alpha = \mathsf {id}$ . For $n \in \mathbb {N}_{\geq 1}$ , $\alpha \in [1/(n+1), 1/n]$ , and $m \in \mathbb {N}_{\geq 1}$ , define

$$\begin{align*}\left(f^{\alpha}(\Phi)\right)_m = \begin{cases} \Phi_m, & m < n\\ \sqrt{\beta}\,\, \Phi_m, & m = n\\ 0, & n < m < 2n\\ \sqrt{1-\beta}\,\, \Phi_m, & m = 2n\\ \Phi_{m/2}, & m> 2n\text{ and }m\text{ even}\\ 0, & m > 2n\text{ and }m\text{ odd,} \end{cases} \end{align*}$$

where $\beta = (1/n - \alpha ) \times (1/n - 1/(n+1))$ . By inspection, we see that $f^\alpha $ gives a homotopy between the identity and $\chi ^2$ , using that $\mathsf {V}(d)$ has the direct limit topology.

We now consider the problem of assigning an approximate cocycle to an approximate local trivialization.

Although we can associate a $3\varepsilon $ -approximate cocycle to every $\varepsilon $ -approximate local trivialization, this choice is not canonical, as an approximate local trivialization can have many distinct witnesses. Nonetheless, the following result says that any two witnesses cannot be too far apart.

The following result gives conditions under which there is a canonical approximate cocycle associated to an approximate local trivialization.

We now give sufficient conditions for an approximate local trivialization to be nondegenerate.

We conclude by remarking that Lemma 4.14 implies that, if $\varepsilon \leq 1$ , we have a canonical map (that is independent of any arbitrary choices)

$$\begin{align*}\mathsf{w} : \mathsf{T}_{\varepsilon}\left( \mathcal{U}; d \right) \to \mathsf{Z}^{1}_{3\varepsilon}\left( \mathcal{U}; O(d) \right) \end{align*}$$

given by taking the best witness, as in Lemma 4.13.

4.2 Local trivializations and classifying maps

In this section, we give a construction that, given an approximate local trivialization, returns an approximate classifying map. We also observe that this construction behaves well with respect to homotopies between approximate local trivializations.

Recall from Section 2.1.2 the definition of the map $\mathsf {Proj} : \mathsf {V}(d) \to \mathsf {Gr}(d)$ .

Construction 4.15. Let $\mathcal {U}$ be a countable open cover of a paracompact topological space B and let $\varphi $ be a partition of unity subordinate to $\mathcal {U}$ . Let $\Phi $ be an approximate local trivialization subordinate to $\mathcal {U}$ . Define a map $\mathsf {av}^\varphi (\Phi ) : B \to \mathbb {R}^{\infty \times \infty }$ by

$$\begin{align*}\mathsf{av}^\varphi(\Phi)(y) = \sum_{i \in I} \varphi_i(y) \mathsf{Proj}(\Phi_i(y)).\\[-46pt] \end{align*}$$

$\vartriangleleft $

Note that the map $\mathsf {av}^\varphi (\Phi )$ is continuous.

The following is clear.

4.3 Cocycles and classifying maps

In this section, we relate approximate cocycles to approximate classifying maps in a way that is independent of any partition of unity and of any enumeration of the sets in the open cover the cocycle is subordinate to (Theorem 4.21). We also study the action of refinements on approximate cohomology.

Let B be a paracompact topological space and let $\mathcal {U} = \{U_i\}_{i \in I}$ be a countable open cover. Let $\varphi $ be a partition of unity subordinate to $\mathcal {U}$ and let $\iota : I \to \mathbb {N}_{\geq 1}$ be an injection. Using Lemma 4.3 and Lemma 4.16, we get a map $\mathsf {av}^\varphi \circ \mathsf {triv}^{\varphi ,\iota } : \mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right ) \to \mathsf {Maps}\left ( {B}, {\mathsf {Gr}(d)^{\sqrt {2}\varepsilon }} \right )$ which we compose with the quotient map $\mathsf {Maps}\left ( {B}, {\mathsf {Gr}(d)^{\sqrt {2}\varepsilon }} \right ) \to \left \lbrack {B}, {\mathsf {Gr}(d)^{\sqrt {2}\varepsilon }} \right \rbrack $ to obtain a map

$$\begin{align*}\mathsf{cl}' : \mathsf{Z}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right) \to \left\lbrack {B}, {\mathsf{Gr}(d)^{\sqrt{2}\varepsilon}} \right\rbrack. \end{align*}$$

Together, Lemma 4.4, Lemma 4.17, and Lemma 4.7 imply that this map is independent of the choice of partition of unity $\varphi $ and of injection $\iota $ . The following result says that two approximate cocycles that differ in a $0$ -cochain are sent to the same approximate classifying map by $\mathsf {cl}'$ .

Recall from Section 2.2.4 the notion of refinement of a cover.

Construction 4.19 gives a map $\nu : \mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {V}; O(d) \right ) \to \mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right )$ that descends to a map

$$\begin{align*}\nu : \check{\mathsf{H}}^{1}_{\varepsilon}\left( \mathcal{V}; O(d) \right) \to \check{\mathsf{H}}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right). \end{align*}$$

It is clear that both these maps are $1$ -Lipschitz with respect to $\mathsf {d}_{\mathsf {Z}}$ and with respect to $\mathsf {d}_{\check {\mathsf {H}}}$ .

Proof. We address the case $\varepsilon \in (0,\infty ]$ , the case $\varepsilon = 0$ being similar. We start by defining a $0$ -cochain $\Theta $ subordinate to $\mathcal {U}$ . Given $j \in N(\mathcal {U})$ , let $\Theta _j = \Omega _{\mu (j)\nu (j)}$ if $\mu (j) \neq \nu (j)$ and the identity if $\mu (j) = \nu (j)$ . Let $(jk) \in N(\mathcal {U})$ , and let $y \in U_k \cap U_j$ . To simplify notation in the rest of this proof, let us denote $\Omega _{ab}(y)$ by $\Omega _{ab}$ . We have

$$ \begin{align*} \|\mu(\Omega)_{jk}(y) - (\Theta \cdot \nu(\Omega))_{jk}(y)\| &= \|\Omega_{\mu(j)\mu(k)} - \Omega_{\mu(j)\nu(j)}\Omega_{\nu(j)\nu(k)}\Omega_{\nu(k)\mu(k)}\| \\ &= \|\Omega_{\nu(j)\mu(j)}\Omega_{\mu(j)\mu(k)} - \Omega_{\nu(j)\nu(k)}\Omega_{\nu(k)\mu(k)}\|\\ &\leq \|\Omega_{\nu(j)\mu(j)}\Omega_{\mu(j)\mu(k)} - \Omega_{\nu(j)\mu(k)}\|\\ &\;\;\;\;\;\;+ \|\Omega_{\nu(j)\mu(k)} - \Omega_{\nu(j)\nu(k)}\Omega_{\nu(k)\mu(k)}\|\\ &< 2 \varepsilon, \end{align*} $$

where for the second equality we used the fact that $\Omega _{\nu (j)\mu (j)} = \Omega _{\mu (j)\nu (j)}^{-1}$ combined with the fact that the Frobenius norm is invariant under multiplication by an orthogonal matrix, and for the inequalities we used the triangle inequality and the approximate cocycle condition.

We are now ready to state and prove the main result of this section.

Theorem 4.21. Let B be a paracompact topological space, and let $\mathcal {U}$ be a countable cover of B. Let $\varepsilon \in [0,\infty ]$ . The map $\mathsf {cl}'$ induces a map

$$\begin{align*}\mathsf{cl} : \check{\mathsf{H}}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right) \to \left\lbrack {B}, {\mathsf{Gr}(d)^{\sqrt{2}\varepsilon}} \right\rbrack\end{align*}$$

such that, if $\mathsf {d}_{\check {\mathsf {H}}}\left ( \Omega , \Lambda \right ) < \delta $ in $\check {\mathsf {H}}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right )$ , then $\mathsf {cl}(\Omega )$ and $\mathsf {cl}(\Lambda )$ become equal in $\left \lbrack {B}, {\mathsf {Gr}(d)^{\sqrt {2}(\varepsilon + \delta )}} \right \rbrack $ . Moreover, if $\mu ,\nu : \mathcal {V} \to \mathcal {U}$ are refinements and $\mathcal {V}$ is a countable cover of B, then $\mathsf {cl}(\mu (\Omega ))$ and $\mathsf {cl}(\nu (\Omega ))$ become equal in $\left \lbrack {B}, {\mathsf {Gr}(d)^{2\sqrt {2}\varepsilon }} \right \rbrack $ .

Proof. The map $\mathsf {cl}$ is well defined thanks to Lemma 4.18. For the stability of $\mathsf {cl}$ , note that, using Lemma 4.17, we see that if $\Omega $ and $\Lambda $ are $\varepsilon $ -approximate cocycles subordinate to a countable cover $\mathcal {U} = \{U_i\}_{i \in I}$ , $\varphi $ is a partition of unity subordinate to $\mathcal {U}$ , and $\iota : I \to \mathbb {N}_{\geq 1}$ is an injection, then

$$\begin{align*}\mathsf{d}_{\mathsf{C}}\left( \mathsf{av}^\varphi \circ \mathsf{triv}^{\varphi,\iota}\, (\Omega)\,, \mathsf{av}^\varphi \circ \mathsf{triv}^{\varphi,\iota}\, (\Lambda) \right) \leq \sqrt{2} \mathsf{d}_{\mathsf{Z}}\left( \Omega, \Lambda \right). \end{align*}$$

The stability then follows from Lemma 3.10. Finally, the claim about refinements follows directly from Lemma 4.20.

Note that the map $\mathsf {cl}$ is independent of any choice of partition of unity or enumeration of the cover $\mathcal {U}$ . We conclude with an interesting remark that is not used in the rest of the paper.

Remark 4.22. Let $\mathsf {cov}(B)$ be the category whose objects are the countable covers of a paracompact topological space B and whose morphisms are the refinements. Let $\varepsilon \in [0,\infty ]$ . Construction 4.19 gives a functor $\check {\mathsf {H}}^{1}_{\varepsilon }\left ( -; O(d) \right ) : \mathsf {cov}(B)^{\mathsf {op}} \to \mathbf {Met}$ . We can then define

$$\begin{align*}\check{\mathsf{H}}^{1}_{\varepsilon}\left( B; O(d) \right) = \operatorname*{\mathrm{colim}}_{\mathcal{U} \in \mathsf{cov}(B)} \check{\mathsf{H}}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right), \end{align*}$$

with the caveat that $\check {\mathsf {H}}^{1}_{\varepsilon }\left ( B; O(d) \right )$ may be a pseudo metric space. Theorem 4.21 implies that there is a well-defined map $\mathsf {cl} : \check {\mathsf {H}}^{1}_{\varepsilon }\left ( B; O(d) \right ) \to \left \lbrack {B}, {\mathsf {Gr}(d)^{2\sqrt {2}\varepsilon }} \right \rbrack $ , natural in $\varepsilon \in [0,\infty ]$ .

4.4 Relationship to classical vector bundles

We now relate approximate vector bundles to exact vector bundles, following the intuition that $\varepsilon $ -approximate vector bundles should correspond to true vector bundles as long as $\varepsilon $ is sufficiently small. For this, we use Theorem 4.21. In the case where an approximate cocycle represents a true vector bundle, we study the problem of constructing an exact cocycle that represents the same vector bundle. We also give upper and lower bounds for the distance from an approximate cocycle to an exact cocycle representing the same vector bundle (Theorem 4.27 and Lemma 4.30).

We start by recalling that small thickenings of the Grassmannian embedded in $\mathbb {R}^{\infty \times \infty }$ retract to the Grassmannian. More precisely, if $\varepsilon \leq \sqrt {2}/2$ , there is a map $\pi : \mathsf {Gr}(d)^\varepsilon \to \mathsf {Gr}(d)$ which is a homotopy inverse of the inclusion $\mathsf {Gr}(d) \subseteq \mathsf {Gr}(d)^\varepsilon $ , by Proposition A.12. Let B be a topological space. By postcomposing with $\pi $ , we get an inverse for the natural map $\left \lbrack {B}, {\mathsf {Gr}(d)} \right \rbrack \to \left \lbrack {B}, {\mathsf {Gr}(d)^\varepsilon } \right \rbrack $ which we denote by

$$\begin{align*}\pi_* : \left\lbrack {B}, {\mathsf{Gr}(d)^\varepsilon} \right\rbrack \to \left\lbrack {B}, {\mathsf{Gr}(d)} \right\rbrack. \end{align*}$$

By an abuse of notation, we also let $\pi _* : \mathsf {Maps}\left ( {B}, {\mathsf {Gr}(d)^\varepsilon } \right ) \to \mathsf {Maps}\left ( {B}, {\mathsf {Gr}(d)} \right )$ .

Recall that we constructed a map $\mathsf {cl} : \check {\mathsf {H}}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right ) \to \left \lbrack {B}, {\mathsf {Gr}(d)^{\sqrt {2}\varepsilon }} \right \rbrack $ , so, if $\varepsilon \leq 1/2$ , then any $\varepsilon $ -approximate cocycle $\Omega $ represents a true vector bundle, namely $\pi _*(\mathsf {cl}(\Omega )) \in \left \lbrack {B}, {\mathsf {Gr}(d)} \right \rbrack $ . To summarize, if $\varepsilon \leq 1/2$ , we have defined a map

$$\begin{align*}\pi_* \circ \mathsf{cl} : \check{\mathsf{H}}^{1}_{\varepsilon}\left( \mathcal{U}; O(d) \right) \to \left\lbrack {B}, {\mathsf{Gr}(d)} \right\rbrack. \end{align*}$$

Upper bound

For the rest of this section, we let $\Phi $ be an $\varepsilon $ -approximate local trivialization subordinate to a countable cover $\mathcal {U} = \{U_i\}_{i \in I}$ of a paracompact topological space B, with $\varepsilon \in [0,\infty ]$ ; we let $\varphi $ be a partition of unity subordinate to $\mathcal {U}$ and let $\iota : I \to \mathbb {N}_{\geq 1}$ be an injection. Recall from Lemma 4.16 that

$$\begin{align*}\mathsf{av}^\varphi(\Phi)(y) = \sum_{i \in I} \varphi_i(y) \, \mathsf{Proj}(\Phi_i(y))\end{align*}$$

defines a $\sqrt {2}\varepsilon $ -approximate classifying map $B \to \mathsf {Gr}(d)^{\sqrt {2}\varepsilon }$ . We will make use of results in Appendix A.2.

Lemma 4.23. If $\varepsilon \in (0,\infty ]$ , then for, $i \in I$ and $y \in U_i$ , we have

$$\begin{align*}\left\| \Phi_i(y)\Phi_i(y)^t - \pi_*\left(\mathsf{av}^\varphi(\Phi)(y)\right) \right\| < 2\sqrt{2}\varepsilon \;\;\text{ and }\;\; \left\| \Phi_i(y) - \pi_*\left(\mathsf{av}^\varphi(\Phi)(y)\right)\Phi_i(y)\right\| < 2\sqrt{2}\varepsilon. \end{align*}$$

If $\varepsilon = 0$ , then $\Phi _i(y)\Phi _i(y)^t = \pi _*\left (\mathsf {av}^\varphi (\Phi )(y)\right )$ and $\Phi _i(y) = \pi _*\left (\mathsf {av}^\varphi (\Phi )(y)\right )\Phi _i(y)$ .

Proof. We address the case $\varepsilon \in (0,\infty ]$ , the case $\varepsilon = 0$ being similar. The second inequality is a consequence of the first one and Lemma A.5. For the first inequality, use Lemma A.2 together with Lemma A.6.

Lemma 4.24. Assume that $\varepsilon \leq \sqrt {2}/4$ . For every $i \in I$ and $y \in U_i$ , the matrix $\pi _*\left (\mathsf {av}^\varphi (\Phi )(y)\right )\Phi _i(y)$ has rank d.

Proof. To simplify notation, let us omit from the formulas the evaluations on $y \in U_i$ . It is enough to show that $\left (\pi _*(\mathsf {av}^\varphi (\Phi ))\Phi _i\right )^t\pi _*(\mathsf {av}^\varphi (\Phi ))\Phi _i$ has full rank. Note that

$$ \begin{align*} \left(\pi_*(\mathsf{av}^\varphi(\Phi))\,\Phi_i\right)^t \,\, \pi_*(\mathsf{av}^\varphi(\Phi))\,\,\Phi_i &= \Phi_i^t \,\,\pi_*(\mathsf{av}^\varphi(\Phi))\,\,\pi_*(\mathsf{av}^\varphi(\Phi))\,\,\Phi_i = \Phi_i^t\,\, \pi_*(\mathsf{av}^\varphi(\Phi))\,\,\Phi_i. \end{align*} $$

Using Lemma 4.23 and Lemma A.5, we conclude that

$$\begin{align*}\left\| \Phi_i^t \,\, \pi_*\left(\mathsf{av}^\varphi(\Phi)\right) \,\, \Phi_i - \mathsf{id}\right\| = \| \Phi_i^t \,\, \pi_*(\mathsf{av}^\varphi(\Phi)) \,\, \Phi_i - \Phi_i^t \,\, \Phi_i \,\, \Phi_i^t \,\, \Phi_i\| < 2\sqrt{2}\varepsilon. \end{align*}$$

The result then follows from Lemma A.15, as $2\sqrt {2}\varepsilon \leq 1$ by assumption.

Given an $\varepsilon $ -approximate local trivialization $\Phi $ with $\varepsilon \leq \sqrt {2}/4$ , we now define an exact local trivialization $\Psi $ that represents the same vector bundle. Given $i \in I$ and $y \in U_i$ , we let

$$\begin{align*}\Psi_i(y) := Q\left(\pi_*\left(\mathsf{av}^\varphi(\Phi)(y)\right)\,\,\Phi_i(y)\right), \end{align*}$$

where the map Q is the one of Corollary A.4. By Lemma 4.24 and Corollary A.4, the maps $\Psi _i$ are well defined and continuous. To see that it is an exact local trivialization it suffices to check that, if $y \in U_j \cap U_i$ , then the columns of $\Psi _i(y)$ and $\Psi _j(y)$ span the same subspace of $\mathbb {R}^\infty $ . This is a consequence of the fact that the columns of $\Psi _i(y)$ span the image of $\pi _*(\mathsf {av}^\varphi (\Phi )(y)$ . We also deduce the following.

Lemma 4.25. Let $\varphi $ be a partition of unity subordinate to $\mathcal {U}$ . Then $\mathsf {av}^\varphi (\Psi ) = \pi _*(\mathsf {av}^\varphi (\Phi ))$ .

We now bound the distance between $\Psi $ and $\Phi $ .

Lemma 4.26. Assume that $\varepsilon \leq \sqrt {2}/4$ , then $\mathsf {d}_{\mathsf {T}}\left ( \Phi , \Psi \right ) \leq 4 \sqrt {2}\varepsilon $ .

Proof. To simplify notation, let us omit from the formulas the evaluations on $y \in U_i$ . For every $i \in I$ and $y \in U_i$ , we have

$$ \begin{align*} \left\| \Phi_i - Q\left(\pi_*\left(\mathsf{av}^\varphi(\Phi)\right)\Phi_i\right)\right\| \leq &\,\,\, \| \Phi_i - \pi_*(\mathsf{av}^\varphi(\Phi))\Phi_i\| + \left\| \pi_*(\mathsf{av}^\varphi(\Phi))\Phi_i - Q\left(\pi_*\left(\mathsf{av}^\varphi(\Phi)\right)\Phi_i\right)\right\| \\ < &\,\,\, 2\sqrt{2}\varepsilon + 2\sqrt{2}\varepsilon = 4\sqrt{2} \varepsilon, \end{align*} $$

where we bounded the first summand using Lemma 4.23 and the second summand using Lemma A.8. In order to satisfy the hypotheses of Lemma A.8, we use the same argument as in Lemma 4.24.

Theorem 4.27. Let $\varepsilon \leq \sqrt {2}/4$ and let $\Omega \in \mathsf {Z}^{1}_{\varepsilon }\left ( \mathcal {U}; O(d) \right )$ . There exists $\Lambda \in \mathsf {Z}^{1}_{}\left ( \mathcal {U}; O(d) \right )$ such that $\mathsf {cl}(\Lambda ) = \pi _*(\mathsf {cl}(\Omega ))$ and such that $\mathsf {d}_{\mathsf {Z}}\left ( \Omega , \Lambda \right ) \leq 9\varepsilon $ .

Proof. Let $\varphi $ be a partition of unity subordinate to $\mathcal {U}$ . Since $\varepsilon \leq \sqrt {2}/4 \leq 1/2$ , it follows that $\pi _*(\mathsf {cl}(\Omega ))$ is well defined. By construction, $\pi _*(\mathsf {cl}(\Omega ))$ is the homotopy class of $\pi _* \circ \mathsf {av}^\varphi \circ \mathsf {triv}^{\varphi ,\iota }(\Omega )$ . By Lemma 4.26, there is an exact local trivialization $\Psi $ such that $\mathsf {d}_{\mathsf {T}}\left ( \mathsf {triv}^{\varphi ,\iota }(\Omega ), \Psi \right ) \leq 4\sqrt {2} \varepsilon $ and such that $\mathsf {av}^\varphi (\Psi ) = \pi _* \circ \mathsf {av}^\varphi \circ \mathsf {triv}^{\varphi ,\iota }(\Omega ) $ . Let $\Lambda = \mathsf {w}(\Psi )$ . Then $\Lambda $ is a witness that $\Psi $ is an exact cocycle, so, by Lemma 4.10, we have that $\mathsf {d}_{\mathsf {Z}}\left ( \Omega , \Lambda \right ) \leq \varepsilon + \sqrt {2} \times 4\sqrt {2} \varepsilon = 9 \varepsilon $ . To conclude, note that $\mathsf {cl}(\Lambda ) = [\mathsf {av}^\varphi \circ \mathsf {triv}^{\varphi ,\iota } \circ \mathsf {w}(\Psi )] = [\mathsf {av}^\varphi (\Psi )] = [\pi _* \circ \mathsf {av}^\varphi \circ \mathsf {triv}^{\varphi ,\iota } (\Omega )] = \pi _*(\mathsf {cl}(\Omega ))$ , where in the second equality we used Lemma 4.12 to conclude that $\mathsf {triv}^{\varphi ,\iota }(\mathsf {w}(\Psi ))$ and $\Psi $ are homotopic through $0$ -approximate local trivializations.

5.1 Discrete approximate vector bundles over simplicial complexes

We introduce two notions of discrete approximate vector bundle over a simplicial complex. These notions of discrete approximate vector bundle induce approximate vector bundles over the geometric realization of the simplicial complex.

Since this will be relevant in Section 6, we define discrete approximate cocycles with values in an arbitrary metric group G. Fix a simplicial complex K.

We denote the set of discrete $\varepsilon $ -approximate cocycles on a simplicial complex K with values in G by $\mathsf {DZ}^{1}_{\varepsilon }\left ( K; G \right )$ .

Note that there is a canonical isomorphism of simplicial complexes $N(\mathsf {st}_K) \cong K$ that maps a vertex $i \in K$ to itself.

Note that the map $\mathsf {DZ}^{1}_{\varepsilon }\left ( K; G \right ) \to \mathsf {Z}^{1}_{\varepsilon }\left ( \mathsf {st}_K; G \right )$ is injective. With this in mind, we endow $\mathsf {DZ}^{1}_{\varepsilon }\left ( K; G \right )$ with the metric $\mathsf {d}_{\mathsf {Z}}$ , and interpret the map $\mathsf {DZ}^{1}_{\varepsilon }\left ( K; G \right ) \to \mathsf {Z}^{1}_{\varepsilon }\left ( \mathsf {st}_K; G \right )$ as an embedding of metric spaces.

Denote the set of discrete $\varepsilon $ -approximate local trivializations on a simplicial complex K by $\mathsf {DT}_{\varepsilon }\left ( K; d \right )$ . In this discrete case too, the witness $\Omega $ that $\Phi $ is a discrete $\varepsilon $ -approximate local trivialization is not part of the data of the approximate local trivialization.

Remark 5.6. Using Construction 4.8 we obtain a map $\mathsf {DT}_{\varepsilon }\left ( K; d \right ) \to \mathsf {DZ}^{1}_{3\varepsilon }\left ( K; O(d) \right )$ . If K is finite, this map is algorithmic since the minimization problem

$$\begin{align*}\min_{\Omega \in O(d)} \| \Phi_i \Omega - \Phi_j\| \end{align*}$$

can be solved by using the polar decomposition (Appendix A.2).

The next result guarantees that any vector bundle on a compact triangulable space can be encoded as a discrete approximate cocycle on a sufficiently fine triangulation of the space.

5.2 Reconstruction of vector bundles from finite samples

Let $\delta \geq 0$ and $\ell> 0$ . A function $f : X \to Y$ between metric spaces is a $\boldsymbol {\delta }$ -approximate $\boldsymbol {\ell }$ -Lipschitz map if, for every $x,x' \in X$ , we have $\ell d_X(x,x') + \delta \geq d_Y(f(x), f(x'))$ .

Note that the cover $\{ B(x, \varepsilon )\}_{x \in X}$ is indexed by the elements of X, so the $0$ -simplices of $\check{\mathsf{C}}(X)(\varepsilon )$ consist of the elements of X. As a set, $|\check{\mathsf{C}}(X)(\varepsilon )|$ consists of formal linear combinations

$$\begin{align*}p = \sum_{x \in X} c_x [x] \end{align*}$$

such that $S_p = \{x \in X : c_x> 0\}$ is a finite set with the property that the intersection $\cap _{x \in S_p} B(x,\varepsilon ) \subseteq \mathbb {R}^N$ is nonempty. We will often write $\check{\mathsf{C}}(X)(\varepsilon )$ for the geometric realization $|\check{\mathsf{C}}(X)(\varepsilon )|$ .

Construction 5.9. Let $X \subseteq \mathbb {R}^N$ and let $\varepsilon> 0$ , $\delta \geq 0$ and $\ell> 0$ . Assume that X is finite. Let $Y \subseteq \mathbb {R}^M$ , and let $f : X \to Y$ be a $\delta $ -approximate $\ell $ -Lipschitz map with respect to the distances induced by the Euclidean norm $\|-\|_2$ . Define a continuous map

$$ \begin{align*} \check{\mathsf{C}}(f)(\varepsilon) : \check{\mathsf{C}}(X)(\varepsilon) &\to Y^{2\ell \varepsilon + \delta} \subseteq \mathbb{R}^M\\ \sum_{x \in X} c_x [x] &\mapsto \sum_{x \in X}c_x f(x).\\[-46pt] \end{align*} $$

$\vartriangleleft $

The map $\check{\mathsf{C}}(f)$ is well defined since, if $p = \sum _{x \in X} c_x [x]$ is such that $c_{x_0}> 0$ , then $\|f(z) - f(x_0)\|_2 < \ell 2\varepsilon + \delta $ for every $z \in X$ such that $c_z> 0$ , since, in that case, $\|x_0 - z\| < 2\varepsilon $ , as the balls $B(x_0, \varepsilon )$ and $B(z,\varepsilon )$ must intersect.

The following result is well known; see, for example, [Reference Hatcher26, Corollary 4G.3].

Corollary 5.12. Let $X \subseteq \mathbb {R}^N$ be a finite subset, let $\varepsilon> 0$ and let $\varphi = \{\varphi _x\}_{x\in X}$ be a partition of unity subordinate to $\{ B(x,\varepsilon ) \}_{x \in X}$ . Then, the following map is a homotopy equivalence:

$$ \begin{align*} R^\varphi : X^\varepsilon &\to \check{\mathsf{C}}(X)(\varepsilon) \\ z &\mapsto \sum_{x \in X} \varphi_x(z) [x].\\[-46pt] \end{align*} $$

□

Our reconstruction theorem for vector bundles builds on the following result by Niyogi, Smale and Weinberger, which allows one to recover the homotopy type of a compact manifold smoothly embedded into $\mathbb {R}^N$ from a sufficiently close and dense sample. In the result, $d_H$ denotes the Hausdorff distance.

Proposition 5.13 [Reference Partha Niyogi49, Proposition 7.1].

Let $\mathcal {M} \subseteq \mathbb {R}^N$ be a smoothly embedded compact manifold with $\tau = \operatorname {\mathrm {reach}}(M)> 0$ . Let $P\subseteq \mathbb {R}^N$ such that $d_H(P,\mathcal {M}) < \varepsilon < (3 - \sqrt {8}) \tau $ and let

$$\begin{align*}\alpha \in \left( \frac{( \varepsilon + \tau) - \sqrt{ \varepsilon^2 + \tau^2 - 6\tau \varepsilon}}{2}, \frac{( \varepsilon + \tau) + \sqrt{ \varepsilon^2 + \tau^2 - 6\tau \varepsilon}}{2} \right), \end{align*}$$

which is a nonempty open interval. Then $\mathcal {M} \subseteq P^\alpha $ and the inclusion $\mathcal {M} \to P^\alpha $ is a homotopy equivalence.

We are now ready to prove the reconstruction theorem for vector bundles. Before doing so, we give a short remark about representing vector bundles by Lipschitz maps.