Roughly speaking, I am interested in understanding what (if any) relationships there are between certain obstructions to constructing sections of a bundle of the form $$ P\times_G G/H\to M $$ where $M$ is a (compact) manifold, $G$ is a (compact) Lie group with Lie subgroup $H$ and $P\to M$ is a principal $G$-bundle.
More precisely, I know there is an associated universal bundle $$ EG\times_G G/H\to BG $$
where $EG\to BG$ is the bundle that classifies principal $G$ bundles. I also know that the bundle above admits a Moore-Postnikov tower $\{Z_i\}_{i\in \mathbb{N}}$ such that $Z_{n+1}\to Z_n$ comes from pulling back the fibration $PK_n\to K_n$ (here $K_n=K(\pi_n(G/H),n+1)$ and $PK_n$ is the corresponding path space) by a map $Z_n\to K_n$. Such a map gives rise to a cohomology class $o_{n+1}\in H^{n+1}(Z_n,\pi_n(G/H))$, which give obstructions to lifting constructing a section of $EG\times_G G/H\to BG$ over the various skelta of $BG$.
There is a similar Moore-Postnikov tower $\{W_i\}_{i\in \mathbb{N}}$ for the fibration $$EG\to BG$$ that gives rise to similar obstruction classes $\tilde o_{n+1}\in H^{n+1}(W_n,\pi_n(G))$. Since $EG$ is contractible, I believe that some people might call this a Whitehead tower. At least that is what I believe from looking at the wikipedia page for Postnikov towers.
We have a commutative diagram of fibrations $$\require{AMScd} \begin{CD} EG @>>> BG;\\ @VVV @VVV \\ EG\times_G G/H @>>> BG; \end{CD}$$
where the right hand vertical map is the identity and the other three maps are the other maps are the obvious bundle projections.
According to this post, the Moore-Postnikov towers have nice naturality properties and so this diagram should induce maps $\Phi_n:W_n\to Z_n$. There are also maps $\Psi_n:K(\pi_n(G),n+1)\to K(\pi_n(G/H),n+1)$, induced by the natural projection $G\to G/H$. My naive expectation is that the following maps should fit together in a new commutative diagram (possibly just up to homotopy) of the form
$$\require{AMScd} \begin{CD} W_n @>>> K(\pi_n(G),n+1);\\ @V{\Phi_n}VV @V{\Psi_n}VV \\ Z_n @>>> K(\pi_n(G/H),n+1); \end{CD}$$
where the horizontal maps are the ones giving rise to $o_{n+1}$ and $\tilde o_{n+1}$ above. I further naively believe that if $\alpha:H^{n+1}(W_n,\pi_n(G))\to H^{n+1}(W_n,\pi_n(G/H))$ is the map coming from "changing coefficients" then $\alpha(\tilde o_{n+1})=\Phi^\ast_{n}(o_{n+1})\in H^{n+1}(W_n,\pi_n(G/H))$.
Given all of the above, I have a few questions
- Is all of the above true? If not, is it true with some minor modifications?
- If it is true, can anyone provide a proof or a good reference? I have a feeling that if it is true it is not that hard to show and that the reason I can't do it is that I only know that Moore-Postnikov towers exist and have various nice properties, but don't really know how they are constructed.
FYI: my background is in differential geometry and less in algebraic topology. My understanding of classifying space and Postnikov towers is coming from reading Hatcher's Algebraic topology book over the last several days.
For context, I really want to understand the obstructions $o_n$, but they seem complicated. On the other hand the obstructions $\tilde o_n$ appear to be more tractable (at least in small dimensions where they are related to classical characteristic classes, e.g. Stiefel-Whitney/Pontryagin classes).