- #1
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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
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