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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 $k \ne n+1$.


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. $$ \tilde f(g e) = \varphi_x(g) \tilde f(e) \qquad \forall g \in G,\, e \in E_G \,: p(e) = x $$

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