I'm in the process of trying to understand the equivalence between two different approaches to vector bundles, namely defining them to be smooth surjections $E\to B$ that are locally isomorphic to projections $U_\alpha\times\mathbb{R}^n\to U_\alpha$, and defining them to be collections of transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\to\mathrm{GL}_n(\mathbb{R})$ which satisfy the trivial ($g_{\alpha\alpha}=\mathrm{id}$) and cocycle ($g_{\beta\gamma}g_{\alpha\beta}=g_{\alpha\gamma}$) identities.
Where I'm stuck right now is in trying to prove the following, intuitively true, statement: given an open cover $B=\bigcup U_\alpha$ and two collections of transition functions $g_{\alpha\beta},h_{\alpha\beta}:U_\alpha\cap U_\beta\to\mathrm{GL}_n(\mathbb{R})$ for which the $g_{\alpha\beta}$ are homotopic to the $h_{\alpha\beta}$ via a family of transition functions, the vector bundles they define are isomorphic. To rephrase my hypotheses, I'm saying that we have homotopies $g_{\alpha\beta}^t:(U_\alpha\cap U_\beta)\times I \to \mathrm{GL}_n(\mathbb{R})$ from $g_{\alpha\beta}^0=g_{\alpha\beta}$ to $g_{\alpha\beta}^1=h_{\alpha\beta}$, and for each $t\in I$, the $g_{\alpha\beta}^t$ form a family of transition functions.
I find this to be intuitively true because "vector bundles being isomorphic" to me is a "rigid" property, so that if $E_t$ is the vector bundle defined by the $g_{\alpha\beta}^t$, then if $t=t_0$ were the first point at which the isomorphism class of $E_t$ changes, we should expect some problem to arise there.
So, I'd like to ask for a proof of my claim if it's true, or a counterexample if it is false. If I could ask further, what about the converse? Also, what happens if we remove the requirement that the $g_{\alpha\beta}$ be homotopic to the $h_{\alpha\beta}$ via a family of transition functions, and just require that for each $\alpha,\beta$, they are homotopic?
Thanks in advance for any help you can provide.
Best Answer
Given a «homotopy of transition functions» $(g_{\alpha,\beta}^t)$ you can construct a vector bundle on $B\times [0,1]$ whose restriction to $B\times\{0\}$ and to $B\times\{1\}$ are your vector bundles $E_0$ and $E_1$.
What you want now follows from the homotopy invariance of vector bundles; see, for example, section 4 in the third chapter of the book by Dale Husemoller on Fiber Bundles.