References for classifying spaces and cohomology of classifying spaces

I am interested in understanding classifying spaces and cohomology groups of clasifying spaces to understand characteristic classes as in How do one Introduce characteristic classes question.

Any references are welcome. I have seen Hausemoller's fiber bundles. They have only given little about clasifying space and that Milnor's construction explanation was confusing. Any other references or some short answer saying what this classifying space (of a topological group, lie group, of vector bundles) is welcome.

• You can look at Allen Hatcher's notes on vector bundles and K theory – Thomas Rot Feb 15 '18 at 14:53
• @ThomasRot Yes, I saw that now, he discussing classification of vector bundle (upto homotopy). – user312648 Feb 15 '18 at 15:20
• @ThomasRot I am reading first chapter in that book and they conclude with idea of Universal bundle $E_n\rightarrow G_n$ with the property that all $n$ dimensional bundles over paracompact spaces(we are not on smooth manifolds, we are working only on topological spaces) are obtainable as pullbacks of this single bundle. They say nothing about classification of vector bundles on manifolds, classification of vector bundles on smooth manifolds, classification of principal g bundles on Smooth manifolds. Can you suggest some reference or should I ask it as a separate question. – user312648 Feb 16 '18 at 10:49

I am reading first chapter in that book and they conclude with idea of Universal bundle $E_n\rightarrow G_n$ with the property that all n dimensional bundles over paracompact spaces(we are not on smooth manifolds, we are working only on topological spaces) are obtainable as pullbacks of this single bundle. They say nothing about classification of vector bundles on manifolds, classification of vector bundles on smooth manifolds, classification of principal g bundles on Smooth manifolds. Can you suggest some reference or should I ask it as a separate question.
Since every manifold is paracompact the continuous vector bundles are clearly classified by the maps you mention. Then the question becomes if two smooth vectorbundles are isomorphic as continuous bundles, they are isomorphic as smooth bundles: This is true. The bundle $E_n\rightarrow G_n$ is smooth, and smooth bundles are classified by smooth homotopy classes of maps into $G_n$. But any two (smooth) maps which are homotopic as continuous maps are smoothly homotopic. This follows from the fact that any continuous map can be approximated by a smooth one.
About the principal $G$ bundles: I think there is a universal principal $G$ bundle over $BG$ and any principal $G$ bundle arises as the pullback of this bundle along a classifying map. A reference for this would be Husemöller's Fiber bundles if I remember correctly.
• Ok, Can I see that as follows. Any vector bundle has fiber as $\mathbb{R}^n$ so for this $n$ they have constructed a universal bundle $E_n\rightarrow G_n$ Such that any rank $n$ Real vector bundle is pullback of this bundle. Good.. in case of principal bundle, fiber is a Lie group $G$ (topological group if we are working with topological principal bundle so), so for this $G$ they have constructed a universal principal bundle $EG\rightarrow BG$ such that any principal $G$ bundle is pull back of this bundle..is it the case or you want to correct something. – user312648 Feb 16 '18 at 12:12