So, I'm reading the proof of the Fundamental Product Theorem from Hatcher's Vector Bundles and K-theory, which basically involves thinking hard about the so-called generalized clutching functions, which are simply automorphisms of a vector bundle $E \times S^{1} \to X \times S^{1}$, where $X$ is a compact Hausdorff space and $E$ is a complex vector bundle over $X$.
Since $X$ is compact, there exists a finite open cover $\mathcal{U} = \left\{U_{1},\ldots,U_{k}\right\}$ such that $E|_{U_{i}}$ is trivial and of dimension, say $n_{i}$, for $1 \leq i \leq k$. Then, over each $U_{i}$, an automorphism $f: E \times S^{1} \to E \times S^{1}$ admits a local description as a continuous function $f_{i}: U_{i} \times S^{1} \to GL_{n_{i}}\left(\mathbb{C}\right)$.
However, in the discussion in Hatcher's text, the generalized clutching functions are often taken to be globally described by continuous functions $f: X \times S^{1} \to GL_{n}\left(\mathbb{C}\right)$. For instance, in Proposition 2.7 on pg. 46, linearized clutching functions are taken to be of the form $a\left(x\right)z + b\left(x\right)$, where the dependence on $i$ seems to be supressed.
However, it is in the following passage on pg. 45 that I am able to highlight my confusion most clearly:
In reference to the above quoted passage, my confusion narrows down to these complaints:
- Each $\ell_{i}$ is a map from $X \times S^{1}$ to $GL_{n_{i}}\left(\mathbb{C}\right)$. So, the summation $\ell = \sum_{i}{\varphi_{i} \ell_{i}}$ does not even make sense, unless we are taking direct sum of the matrices.
- Let us suppose $\ell$ is given by the direct sum, so that it is a map from $X \times S^{1}$ to $GL_{N}\left(\mathbb{C}\right)$, where $N = \sum_{i}{n_{i}}$. What do we mean when we say it approximates $f$ over all of $X \times S^{1}$? For $f$ is an automorphism of $E \times S^{1}$, not a map from $X \times S^{1} \to GL_{N}\left(\mathbb{C}\right)$.
I was able to figure it out with help from my professor. This answer here is just in case anyone else has the same question in the future.
It turns out Hatcher isn't claiming that every $f \in \text{Aut}\left(E \times S^{1} \right)$ can be thought of as a map $f: X \times S^{1} \to GL_{n}\left(\mathbb{C}\right)$ at all. I simply misread. However, most of the notational choices rely on the identification of $\text{End}\left(E \times S^{1}\right)$ with $\mathcal{C}\left(S^{1},\text{End}\left(E\right)\right)$, the space of continuous maps from $S^{1}$ to the vector space $\text{End}\left(E\right)$.
Given $f \in \text{End}\left(E \times S^{1}\right)$, we can associate to it a continuous map $$\hat{f}: S^{1} \to \text{End}\left(E\right); z \mapsto f|_{X \times \left\{z\right\}}$$ On the other hand, given a continuous map $f: S^{1} \to \text{End}\left(E\right)$, we can associate to it an endomorphism $$\check{f}: E \times S^{1} \to E \times S^{1}; \check{f}\left(v,z\right) = \left(f\left(z\right)\cdot v, z\right)$$ This allows us to identify the space of endomorphisms $\text{End}\left(E \times S^{1}\right)$ with the space of continuous maps $\mathcal{C}\left(S^{1},\text{End}\left(E\right)\right)$. Furthermore, this identification restricts to an identification between $\text{Aut}\left(E\times S^{1}\right)$ and $\mathcal{C}\left(S^{1},\text{Aut}\left(E\right)\right)$, giving us a workable description of the generalized clutching functions.
Finally, we observe that $\text{End}\left(E \times S^{1}\right) \cong \mathcal{C}\left(S^{1},\text{End}\left(E\right)\right)$ forms a vector space over $\mathbb{C}$ and a module over $\mathcal{C}\left(X \times S^{1}\right)$, which contains the map $X \times S^{1} \to \mathbb{C}; \left(x,z\right) \mapsto z$.
Thus, when we talk of the linear clutching function given by $a\left(x\right) z + b\left(x\right)$, as Hatcher does on pg. 46, $a$ and $b$ are elements of $\text{End}\left(E\right)$, and $a\left(x\right)$, $b\left(x\right)$ are linear maps from $E_{x}$ to itself.
Coming to the quoted paragraph, the $l_{i}\left(x,z\right)$'s do define maps $X_{i} \times S^{1} \to GL_{n_{i}}\left(\mathbb{C}\right)$ and hence also elements of $\text{Aut}\left(E|_{X_{i}} \times S^{1}\right)$. Then, the product $\varphi_{i} l_{i}$ is a well-defined element of $\text{End}\left(E \times S^{1}\right)$, and so the summation $\sum_{i}{\varphi_{i}l_{i}}$ is occurring in the vector space $\text{End}\left(E \times S^{1}\right)$ and all's well with the world. In the paragraph immediately preceding the quoted paragraph, Hatcher equips $\text{End}\left(E \times S^{1}\right)$ with the norm $$ ||f|| = \text{sup}_{z \in S^{1}}||f\left(z\right)\left(x\right)||$$ where $f\left(z\right)\left(x\right)$ is a linear operator on $E_{x} \times \left\{z\right\}$ and $|| \cdot ||$ is the standard operator norm. Now, the claim that $\sum_{i}{\varphi_{i}l_{i}}$ approximates $f$ makes complete sense.