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Suppose a space $X'$ is obtained from $X$ by attaching an $i$-cell, i.e., $$X' = X \cup_\phi e^i$$ where $\phi: \partial e^i \to X$ is the attaching map. Let $G$ be a structure group, and $P$ be a principal $G$-bundle on $X'$, classified by the map $f: X \to BG$. The bundle $P$ can be assembled from its restrictions to $X$ and $e^i$, together with descent data; there is a pushout square

$$ \require{AMScd} \begin{CD} \partial e^i \times G @>>> e^i \times G \\ @VV{\tilde{\phi}}V @VVV\\ P|_X @>>> P. \end{CD} $$

The map $\tilde{\phi}$ is a bundle map over $\phi$, and is in principle determined by the classifying map $f$.

What is the attaching map $\tilde{\phi}$ in terms of $f$?

In other words, what's the map $$\tilde{\phi}: G \times \partial e^i \cong EG \times_{BG} \partial e^i \xrightarrow{1 \times \phi} EG \times_{BG} X = P|_X?$$

From this perspective, we see that choices of isomorphisms are involved.

If it helps, restrict to the case where $X$ is contractible. Then $P|_X \cong X \times G$, and $\tilde{\phi}$ is determined by a map $$\partial e^i \times G \to G.$$ If the homotopy class of $f$ is known, can we write down an equation for the homotopy class of this map?

Here is a related question.

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  • $\begingroup$ I don't think $\bar\phi$ is going to have a nice description in terms of $f$. For example it exists given $P$ without assuming a particular classifying map is chosen for it. Moreover, it depends more on a particular choice of null-homotopy for $f|_{e^i}$ rather than the map $f$ itself, and this equivalent here to a choice of trivialisation $P|_{e^i}\cong e^i\times G$, $\endgroup$ – Tyrone Apr 17 '19 at 16:34
  • $\begingroup$ @Tyrone Yes, I was worried that this might be the case. So if there's an answer, the phrase "in terms of $f$" must be interpreted very liberally. Since there are choices of isomorphisms involved, I was hoping that the ambiguity can be packaged in the "homotopy class" of the transition function, but this might not be correct. $\endgroup$ – JHF Apr 17 '19 at 17:09

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