# Classifying maps and transition functions

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.

• 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 $P|_{e^i}\cong e^i\times G$, – Tyrone Apr 17 '19 at 16:34
• @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. – JHF Apr 17 '19 at 17:09