**RECAP on classification of bundles**

We want to classify $G$-principal bundles over $X$ (smooth manifold, G compact Lie). These are in 1-1 correspondence with homotopy classes of maps $[X,BG]$ (where $EG\to BG$ is the universal bundle as usual). If $BG \simeq K(\pi,n)$ then it's easy: $$[X, BG]\leftrightarrow H^n(X,\pi),$$ therefore there is a cohomology class that gives us the classification (e.g. the 1st Chern class for the frame bundles of complex line bundles). In general, $BG$ can be more complicated, in any case $BG$ has a Postnikov tower which induces a factorization of the classifying map $f\in [X, BG]$ in $(f_i)_i$ $\require{AMScd}$ \begin{CD} \vdots@. \vdots\\ @| @VVV \\ X@>>f_2> P_2(BG)\\ @| @VVp_2V \\ X @>>f_1> P_1(BG)@.\simeq K(\pi_1(BG),1) \end{CD}

The homotopy type of $f$ is given by the homotopy type of the $f_i$s. Since $P_1(BG)\simeq K(\pi_1(BG),1)$ $f_1$ is given by a cohomology class in $H^1(X, \pi_1(BG))$. However, not any choice of $f_2$ works, because it must lift $f_1$. From this answer* I learned that there is a Cartesian diagram $\require{AMScd}$ \begin{CD} X@>>f_2> P_{2}(BG) @>>> K(\pi_0G,1)\\ @| @VVp_2V @VVV\\ X @>>f_1> P_1(BG) @>>> K(\pi_2 G,3)_{h\pi_0G} \end{CD}

1)Explanation/references for this? I was expecting the second column to be something like $K(\pi_2(BG),2)\to K(\pi_1(BG),1)$, it reminds me of principal fibrations.

2)How to see that these lifts are parametrized by $H^{2}(X,\pi_{2}(BG))$ cohomology with local coefficients twisted by $f_1\in H^1(X,\pi_1(BG))$? Obstruction theory tells the necessary conditions to lift $f_1$ but not how many lifts there are.

In the end we get that the principal bundle is classified by $f_1\sim \alpha_1 \in H^1(X,\pi_1(BG))$ and a sequence of cohomology classes $\alpha_k \in H^{k}(X, \pi_k(BG)) $ in the cohomology with local coefficients twisted by $\alpha_1$.

3) How to compute them? Is there any example for say $G=O(2)$? Any link with invariant polynomials in $\mathfrak{g}$ or the Weyl algebra of $\mathfrak{g}$?

This is essentially Denis Nardin's answer. In his comment Nardin, says another interesting thing if $G=O(n)$, then $\alpha_1 = 0 $ iff the bundle is orientable, $\alpha_1, \alpha_2 = 0$ iff the bundle is spin and so on climbing the Whitehead tower of $O(n)$ $$O(n)\leftarrow SO(n)\leftarrow Spin(n)\leftarrow String(n)\leftarrow ...$$

4)Is this true for any Whitehead tower of groups? Does this implicitly say that the Postnikov tower of $SO(n)$ $(Spin(n))$ is the one of $O(n)$ without the first (second) term?

**BG as a twisted product**

If $\pi_1(BG)$acts on $\pi_{n+1}(BG)$ trivially then

- the Postnikov tower gives us an expression for $BG$ in terms of a twisted product of Eilenberg-MacLane spaces $BG \simeq K(\pi_1(BG),1)\times_{k_1} K(\pi_2(BG),2)\times_{k_2} \dots$
- There is no need of local coefficients for the $\alpha_k$ above.

In the same answer*, Mark Grant says that in the case of $G=O(2)$:

there is a fibration $$ K(\mathbb{Z},2)\to E\mathbb{Z}/2\times_{\mathbb{Z}/2} K(\mathbb{Z},2)\to B\mathbb{Z}/2 $$ given by the twisting of $w_1$ on the universal $SO(2)$ bundle, and this fibration agrees up to homotopy with the fibration $$BSO(2)\to BO(2)\to BO(1).$$

Question:

5) Can you explain this in the more general setting of a fibration $F\to E\to B$? Also I do not understand $E\mathbb{Z}/2\times_{\mathbb{Z}/2} K(\mathbb{Z},2)$, what is the $\mathbb{Z}/2$ action on $K(\mathbb{Z},2)$? Grant says $w_1$ is involved but I cannot imagine how. (I know that in general $w_1 \in H^1(X, \pi_1(BO(2))$ gives me an action of $\pi_1(X)$ on $\pi_n(BO(2))$). How does this relate to the Whitehead tower above?

*Classification of $O(2)$-bundles in terms of characteristic classes

nota principal fibration, so local coefficients appear $\endgroup$ – Denis Nardin Apr 14 at 15:19