TITLE: Vector bundle transitions and Čech cohomology
QUESTION [24 upvotes]: I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ (for the sheaf of functions with values in $GL(n)$). Could somebody provide a precise statement with an explanation or a reference?
Thank you!

REPLY [14 votes]: As has already been said, there is a bijection betweeen $H^1(M, GL(n, \mathcal{C}^{\infty}_M))$ and the set of all isomorphism classes of smooth rank $n$ vector bundles on $M$. Lee's Introduction to Smooth Manifolds (second edition) is a good book in general, but for your purposes it has some very helpful exercises. If you really want to understand the bijection, I suggest you go through the following exercises:

$10$-$5$: This shows that the transition functions for a smooth rank $n$ vector bundle on $M$ define a Čech $1$-cocycle with values in $GL(n, \mathcal{C}^{\infty}_M)$.
$10$-$6$: This shows that any such cocycle defines a smooth rank $n$ vector bundle on $M$.
$10$-$12$: This shows the relationship between two Čech $1$-cocycles which define isomorphic smooth rank $n$ vector bundles on $M$. 

One would hope that the relationship obtained in Exercise $10$-$12$ is 'cohomologous'. However, we have to be careful because for $n > 1$, $GL(n, \mathcal{C}^{\infty}_M)$ is a sheaf of non-abelian groups; in particular, what does it mean to be cohomologous in this setting? Fortunately, Brylinski's Loop Spaces, Characteristic Classes and Geometric Quantization defines what it means (see Chapter $4$, Section $4.1$, Equation $4$-$2$); note, this is just for the case of $1$-cocycles, but that is all you need. Thankfully, the result you get from Exercise $10$-$12$ tells you that two cocycles defining isomorphic smooth rank $n$ vector bundles are cohomologous. In summary:

For a given cohomology class in $H^1(M, GL(n, \mathcal{C}^{\infty}_M))$, choose a cocycle representative $(\tau_{ij})$ and construct (via Exercise $10$-$6$) the smooth rank $n$ vector bundle $E$ on $M$. Then the map $[(\tau_{ij})] \mapsto [E]$, where square brackets indicate the corresponding cohomology and isomorphism classes respectively, is well-defined and is in fact a bijection between $H^1(M, GL(n, \mathcal{C}^{\infty}_M))$ and the set of all isomorphism classes of smooth rank $n$ vector bundles on $M$.


You can still appreciate what is happening if you don't do the exercises, but I recommend you get a copy of Lee's book and at least look at them. The corresponding exercises in the first edition are $5$-$3$, $5$-$4$, and $5$-$5$.