morphisms of principal bundles with different structure groups

Question

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram:

\begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G & \overset{\tilde f}{\longrightarrow} & E_H \\ \Big\downarrow && \Big\downarrow \\ X & \overset{f}{\longrightarrow} & Y \\ \end{array} \end{equation}

where $E_G \to X$ is a principal $G$-bundle and $E_H \to Y$ is a principal $H$-bundle. The question is: for which "reasonable" conditions there exists a map $\tilde f$ s.t.

1) the diagram above is commutative

2) $\tilde f$ is "compatible" with the principal bundle structure, i.e.: there exists a continuous homomorphism $\varphi \,: G \to H$ s.t. $$ \tilde f(g e) = \varphi(g) \tilde f(e) \qquad \forall e \in E_G,\, g \in G $$

I think that my previous question: extending maps from spaces to their whitehead towers provides a reasonable condition for 1) to be true: $E_G$ and $E_H$ are $n+1$-connected, $X$ and $Y$ are $n$-connected, and $\pi_k(E_G) \to \pi_k(X)$ ($\pi_k(E_H) \to \pi_k(Y)$) are isomorphisms for $k \ne n+1$.

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EDIT a weaker statement than 2) is probabily more appropriate: $\varphi$ depends continuously on the fiber: ($p \: E_G \to X$)

2') $\tilde f$ is "compatible" with the principal bundle structure, i.e.: there exists a continuous assignement of a homomorphism $\varphi_x \,: G \to H$, $x \in X$ s.t. $$ \tilde f(g e) = \varphi_x(g) \tilde f(e) \qquad \forall g \in G,\, e \in E_G \,: p(e) = x $$

Answer 1

I'm not sure what reasonable conditions you are looking for, but hopefully this helps.

When discussing homomorphisms of principal bundles with different structure groups, you should fix the (Lie or topological) group homomorphism $\varphi: G \to H$ as part of your data.

If you check Kolar Michor Slovak's Natural Operations book section III.10 you might find some useful details. For example, they show that such a homomorphism of principal bundles factors as an extension of structure group from $G$ to $H$ followed by a pullback square of principal $H$-bundles. Indeed, we can see that the problem amounts to asking if there exists an isomorphism of principal $H$-bundles $E_G \times_G H \cong f^* E_H$.

Another way of looking at it is that the group homomorphism $\varphi: G \to H$ gives a map $B\varphi: BG \to BH$ between the corresponding classifying spaces. Let $\gamma: X \to BG$ be the classifying map for your $G$-bundle, and $\eta: Y \to BH$ the classifying map for your $H$-bundle. So your question amounts to asking if $\eta \circ f$ is homotopy equivalent to $B\varphi \circ \gamma$.