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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 $EGto BG$ 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 $fin [X, BG]$ in $(f_i)_i$
$requireAMScd$
beginCD
vdots@. vdots\
@| @VVV \
X@>>f_2> P_2(BG)\
@| @VVp_2V \
X @>>f_1> P_1(BG)@.simeq K(pi_1(BG),1)
endCD

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
$requireAMScd$
beginCD
X@>>f_2> P_2(BG) @>>> K(pi_0G,1)\
@| @VVp_2V @VVV\
X @>>f_1> P_1(BG) @>>> K(pi_2 G,3)_hpi_0G
endCD

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 $f_1in H^1(X,pi_1(BG))$? 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
$f_1sim alpha_1 in H^1(X,pi_1(BG))$ 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 $mathfrakg$ or the Weyl algebra of $mathfrakg$?

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

1. the Postnikov tower gives us an expression for $BG$ in terms of a twisted product of Eilenberg-MacLane spaces $BG simeq K(pi_1(BG),1)times_k_1 K(pi_2(BG),2)times_k_2 dots$
   


2. 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
$$
K(mathbbZ,2)to EmathbbZ/2times_mathbbZ/2 K(mathbbZ,2)to BmathbbZ/2
$$
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 $Fto Eto B$? Also I do not understand $EmathbbZ/2times_mathbbZ/2 K(mathbbZ,2)$, what is the $mathbbZ/2$ action on $K(mathbbZ,2)$? 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

I'll try to answer question 1. Unfortunately, I know of no especially convenient reference for the case of understanding general Postnikov towers; however, everything I say below is "well known".

In a Postikov tower for $X$, the map $p=p_ncolon P_nto P_n-1$ is, by definition, a fibration with fiber equivalent to $K(A,n)$, where $A=pi_n X$. The idea is that there exists an (essentially unique) homotopy pullback square
$requireAMScd$
beginCD
P_n @>>> E_F\
@VpVV @VVqV\
P_n-1 @>>> B_F
endCD
where $q$ is the universal example of a fibration with fiber equivalent to $F:=K(A,n)$. This universal fibration is rarely ever a principal bundle.

Here is the formula for $q$: Let $defAutmathrmAutdefMapmathrmMap$ $Aut(F)subseteq Map(F,F)$ be the union of all components of the mapping space which contain homotopy equivalences. Then $Aut(F)$ is a topological monoid which is "grouplike" (i.e., $pi_0Aut(F)$ is a group), and which acts on $F$. Then the map
$$ F_hAut(F) to (*)_hAut(F)=BAut(F)$$
is the universal $F$-fibration.

For $F=K(A,n)$, it turns out you can identify $Aut(F)$ very explicitly. (I'm going to assume $ngeq 2$ here.) The construction of Eilenberg-MacLane $Amapsto K(A,n)$ spaces lifts to a functor
$$(textabelian groups)to (texttopological abelian groups).$$
Therefore we get a topological group $K(A,n)rtimes Aut(A)$ acting on $K(A,n)$ (the $K(A,n)$ acts on itself by left-translation, and $Aut(A)$ is the discrete automorphism group of $A$.) It turns out (by a computation) that this group is equivalent (up to homotopy) to the topological monoid $Aut(K(A,n))$, so
$$B_K(A,n) approx BAut(K(A,n)) approx Bbigl( K(A,n)rtimes Aut(A)bigr) approx BK(A,n)_hAut(A)approx K(A,n+1)_hAut(A),$$
and
$$E_K(A,n) approx K(A,n)_hbigl(K(A,n)rtimesAut(A)bigr) approx (*)_hAut(A) approx BAut(A).$$
So the desired pullback square has the form
beginCD
P_n @>>> BAut(A)\
@VpVV @VVqV\
P_n-1 @>>> K(A,n+1)_hAut(A)
endCD
In practice, when you have a Postnikov tower of a space like $BG$, the action of the fundamental group $pi_1 BG=pi_0G$ on $A=pi_nBG$ determines a homomorphism $pi_0Gto Aut(A)$, and you can "restrict" along this homomorphism to get a pullback square
beginCD
P_n @>>> Bpi_0G\
@VpVV @VVqV\
P_n-1 @>>> K(A,n+1)_hpi_0G
endCD