In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.



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> $H^2(M,\mathbb{Z})$ is the group of isomorphism classes of line bundles over $M$.
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I guess they mean there is a natural isomorphism. 


Can some one give a rough idea of what obvious second cohomology class we can think of given a line bundle over $M$ and what line bundle can we think of given an arbitrary second cohomology class. 


Intuitive comments are also welcome.


I am familiar (not the proof details) with following result:



> 
> If $G$ is a group and $M$ is a $G$-module, then the $H^2(G, A)$ is in one-one correspondence with the set of
>  equivalence classes of extensions $E$ of $M$ by $G$, in which the action of $G$ on $M$ induced by conjugation in $E$ is the same as the action defined by the $G$- module $M$.
> 
> 
> 


I am expecting some intuitive explanation that looks similar to this.



Classification of line bundles by second cohomology of a manifold
I think $H^2(M;\mathbb{Z})$ cannot mean the de Rham cohomology group. The coefficients are wrong. 


Anyway: $\mathbb{CP}^\infty$ is an amazing space. It is both a model for $K(\mathbb{Z},2)$ and a model of $BU(1)$. Homotopy classes into $K(\mathbb Z,2)$ is in bijection with $H^2(M;\mathbb{Z})$ (By pulling back the fundamental class) and homotopy classes into $BU(1)$ classify (complex) line bundles (by pulling back the tautological bundle). This works well with natural group structures. This gives the required isomorphism.


The cohomology class in $H^2(M;\mathbb{Z})$ corresponding to the complex line bundle is the first Chern class. 


I can recommend Milnor Stasheff "characteristic classes" for the classification of the line bundles. I can recommend Hatcher for the classification of cohomology in terms of homotopy classes into $K(\mathbb Z,n)$. 


