Let G be an infinite Lie group. Let mor(P, Q) be the set of all smooth bundle maps between two principal G-bundles P and Q. Similarly, Let mor(F, L) be the set of all smooth bundle maps between two frame bundles F and L with structure group G. How can we relate mor(P, Q) and mor(F, L)?
Frame bundles are induced by principle bundles with orhogonal representtion of structure group. In bundle theory there is a classification and universla space so every morphism of bundles are induced bymorpisms between classification spaces and universal spaces.
Thanks Cenap Özel, It's still not clear to me, if you please give more explanation how to get a map f from mor(P, Q) to mor(F, L). I have a vague idea by using a map from G of the principals to GL(n) of the frame bundle and then use this map to pass from mor(P, Q) to mor(F, L). Correct me if I'm wrong.
all bundle maps are induced from the maps between classifying BG and universal space EG of G where BG=EG/G th ere is a commutative diagram
please look at Husemoller Fiber bundles book It is basic boom for this subject.
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