Introduction
In this chapter we draw two direct consequences of the Serre spectral sequence (Example 18.2). Recall the homological and cohomological Serre spectral sequences associated to a fibration \(F \to E \to B\) with \(\pi_1(B) = 0\) and \(\pi_0(F) = 0\text{:}\)
\begin{equation*}
E^2_{p,q} = H_p\bigl(B;\, H_q(F)\bigr) \;\Longrightarrow\; H_{p+q}(E),
\end{equation*}
\begin{equation*}
E_2^{p,q} = H^p\bigl(B;\, H^q(F)\bigr) \;\Longrightarrow\; H^{p+q}(E).
\end{equation*}
We give two applications:
-
a new proof of the Hurewicz theorem (Theorem 14.2) using the homological Serre sseq applied to the path-space fibration;
-
the Gysin and Wang long exact sequences, available when one of the spaces in the fibration is a sphere.
