Classification

$\pi_1(B)$ acts on the fiber by self-homotopy equivalences, i.e. maps

\begin{align*} \pi_1(B) \to Aut(K(\pi, n)) \cong Aut(\pi), \end{align*}

where the second equivalence is just a consequence of the simplicity of these spaces, i.e. its single homotopy group.

Note: usually we are assuming that this action is trivial; otherwise, accounting for it makes things more complicated.

Assuming that $\pi_1(B) = 0$, or equivalently that the above action is trivial, then isomorphism classes of these bundles correspond to certain cohomology elements. In particular, we have

\begin{align*} \left\{{\text{isomorphism classes of bundles } K(\pi, n)\to E\to B}\right\} \iff H^{n+1}(B; \pi) \end{align*}

In general, the right hand side of this correspondence is referred to as cohomology “with twisted coefficients” in the “bundle of groups” $\pi \to G \to B$.

Why?

Every $K(\pi, n)$ fibration is a pullback of the universal one, i.e.

\begin{align*} \begin{CD} @. @. \Omega K(\pi, n+1) = K(\pi, n)\\ @. @. @VVV \\ K(\pi, n) @>>> E = f^* PK(\pi, n+1) @>>> PK(\pi, n+1) \simeq {\operatorname{pt}}\\ @. @V{p}VV @V{p'}VV \\ @. B @>{f}>> K(\pi, n+1) \\ \end{CD} \end{align*}

and so we can think of the universal bundle as the “classifying space for the $K(\pi, n)$” bundle here.

We can also explicitly construct $f^* E = \left\{{x, (p^{-1}\circ f)(x)}\right\}$.

The universal bundle $K(\pi, n) \to PK(\pi, n+1) \to K(\pi, n+1)$ gives rise to a canonical element in

\begin{align*} H^{n+1}(K(\pi, n+1); \pi) \cong \hom(H_{n+1} (K(\pi, n+1), \pi) = \mathop{\mathrm{Hom}}(\pi, \pi) = \mathop{\mathrm{Aut}}(\pi) \end{align*}

by the universal coefficient theorem, where $\operatorname{Ext} $ disappears since $H_{<n+1}K(\pi, n) = 0$.

The group $Aut(\pi)$ a natural identity element, namely the identity automorphism, and so we define the corresponding element in cohomology to be the characteristic class of the bundle.

Thus, given any $f$ fitting into the following situation

\begin{align*} \begin{CD} K(\pi, n) @>>> PK(\pi, n+1) \\ @. @VVV \\ X @>{f}>> K(\pi, n+1) \end{CD} \end{align*}

we can form $f^*C \in H^{n+1}(X; \pi)$.

Remark: Note that this gives us an alternative definition of cohomology in terms of homotopy, using the correspondence \begin{align*}[X, K(\pi, n)] \cong H^n(X; \pi)\end{align*} .

Claim: This correspondence yields the desired isomorphism.

2021-04-26_Research_Notes_4-18-2018

Classification