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Section 22.2 Whitehead Towers
There is a dual construction that kills homotopy groups
below a given degree.
Theorem 22.2 . Whitehead cover.
Let \(X\) be a connected CW complex. There exists a sequence of fibrations
\begin{equation*}
\cdots \;\to\; X_{>n+1} \;\to\; X_{>n} \;\to\; X_{>n-1} \;\to\; \cdots \;\to\; X_{>1} \;\to\; X_{>0} = X
\end{equation*}
such that
\(X_{>n}\) is \(n\) -connected;
\(\pi_q(X_{>n}) \cong \pi_q(X)\) for \(q > n\text{;}\)
the fibre of \(X_{>n} \to X_{>n-1}\) is a \(K(\pi_n X, n-1)\text{.}\)
Construction: take
\(X \to X_{\leq n-1}\text{,}\) the
\((n-1)\) st Postnikov section, convert it to a fibration, and let
\(X_{>n-1}\) be its homotopy fibre. The space
\(X_{>n}\) is called the
\(n\) -connected cover of
\(X\text{.}\) Item (c) follows from the long exact sequence in homotopy of
\(X_{>n} \to X_{>n-1} \to K(\pi_n X, n)\text{.}\)
