Editing G-structure on a manifold (section)

=== Definition ===
In the following, let <math>X</math> be a [[topological space]], <math>G, H</math> topological groups and a group homomorphism <math>\phi\colon H \to G</math>.

==== In terms of concrete bundles ====
Given a principal <math>G</math>-bundle <math>P</math> over <math>X</math>, a ''reduction of the structure group'' (from <math>G</math> to <math>H</math>) is a ''<math>H</math>''-bundle <math>Q</math> and an isomorphism <math>\phi_Q\colon Q \times_H G \to P</math> of the [[associated bundle]] to the original bundle.

==== In terms of classifying spaces ====
Given a map <math>\pi\colon X \to BG</math>, where <math>BG</math> is the [[classifying space]] for <math>G</math>-bundles, a ''reduction of the structure group'' is a map <math>\pi_Q\colon X \to BH</math> and a homotopy <math>\phi_Q\colon B\phi \circ \pi_Q \to \pi</math>.