2021-04-26_Research_Notes_4-18-2018

We have the following situation: a fiber bundle

\begin{align*} K(\pi, n) \to E \to B \end{align*}

Where we might assume that \(B\) is connected for simplicity.

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.

#undergraduate