When can one reduce the structure group?

[quasar987]

I know of the theorem that says that in any fiber bundle with structure group G having finitely many connected component, the structure group can be reduced to a maximal compact subgroup of G.

But here I am reading a thesis in which the author says: "[Since U(n,n)] is homotopic to its maximal compact subgroup U(n) x U(n), the structure group can be reduced to U(n) x U(n)."

The context here is a vector bundle with structure group G=U(n,n).

So I am wondering, is there a result about reduction of str. groups of vector bundles that says something like "if H<G has the same homotopy type as G, then we can reduce to H" ??

Thx

 

[zhentil]

Yes, it's true. Trivialize the bundle locally with associated transition maps to the structure group. Choose a homotopy between each of those maps and a map to H. This yields a bundle homotopy, hence an isomorphism.

 

[quasar987]

So it's true for fiber bundles in general, and withouth restriction on the number of connected component to G?

(P.S. wikipedia doesn't seem to contain an article including the words 'bundle homotopy'. Do know of another name for the thing?)

 

[zhentil]

I'm not sure what you mean by fibre bundles in general here. For a general fibre bundle, the structure group is the diffeomorphism group of the fiber. This is an infinite dimensional Lie group, but not what you have in mind, I would guess.

A bundle homotopy between two vector bundles E_0 and E_1 is a vector bundle over M x [0,1] such that if one considers the inclusion i_s : M -> M x {s}, the pullback of the bundle satisfies i_s^*(E) =: E(s), with E(0)=E_0 and E(1)=E_1. Homotopic vector bundles are isomorphic.

 

[quasar987]

Oh, I see! Thanks. :)

 

[quasar987]

I carried out the "details", and it seems to me that this works only if H is a deformation retract of G. But perhaps I did not grasp the full power of your idea?

 

[zhentil]

In this context, it only works if H is a deformation retract of G. For the "full power," I think one has to go with either Cech cohomology or classifying spaces. It remains true, however.

 

[quasar987]

Ok, interresting. Any reference?

 

[zhentil]

Steenrod's "The Topology of Fibre Bundles" is the old standby.