principal bundle morphism preserves fundamental group

Question

Two related questions.

1. What is the morphism for principal bundles?
2. Does it "preserve" fundamental groups?

Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to interpret that in this case. I guess that there should be some kind of compatibility condition between the principal bundle morphism and the structure group G.

(I need all this to prove that the hopf fibration is not trivial.)

Answer 1

A morphism $f: P \to Q$ is just a $G$-equivariant fiber bundle map (i.e. if $p_1:P \to B$ and $p_2: Q \to B$ are the bundles, then $p_1 = p_2 \circ f$).

It turns out that all principal $G$ bundle morphisms are isomorphisms. See here for an elementary proof. Thus, in particular, $f_* : \pi_1(P) \to \pi_1(Q)$ is an isomorphism. Is this the sense in which $f$ "preserves fundamental groups" that you were asking about?