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Section 18.6 A Small Computation with the Serre Spectral Sequence

We close the chapter with a sample computation illustrating the formalism above. The Serre spectral sequence of a fibration is constructed in detail in a later chapter; for now we use it as a black box.

Example 18.9. \(H_1(\Omega B) \cong H_2(B)\) for \(B\) simply connected.

Let \(B\) be a simply connected pointed space. The path space fibration
\begin{equation*} \Omega B \;\longrightarrow\; PB \;\xrightarrow{\;\mathrm{ev}_1\;}\; B \end{equation*}
has total space the contractible path space \(PB \simeq *\text{,}\) so \(H_n(PB) = \mathbb{Z}\) for \(n = 0\) and \(0\) otherwise. The Serre spectral sequence of this fibration is
\begin{equation*} E^2_{p,q} \;=\; H_p\bigl(B;\, H_q(\Omega B)\bigr) \;\Longrightarrow\; H_{p+q}(PB) \;\cong\; \begin{cases} \mathbb{Z} & p+q = 0, \\ 0 & p+q \neq 0. \end{cases} \end{equation*}
Pictorially the \(E^2\) page in the relevant range looks like:
The bottom row \(E^2_{p,0} = H_p(B; H_0(\Omega B)) = H_p(B)\) uses that \(\Omega B\) is path-connected, and the leftmost column \(E^2_{0,q} = H_0(B; H_q(\Omega B)) = H_q(\Omega B)\) uses that \(B\) is path-connected. The zero in position \((1, 0)\) is \(H_1(B) = 0\text{,}\) since \(B\) is simply connected.
The diagonal \(p + q = 1\) must vanish on the \(E^\infty\) page, since the abutment is concentrated on the diagonal \(p + q = 0\text{.}\) The only nonzero entry on \(p + q = 1\) at \(E^2\) is \(E^2_{0,1} = H_1(\Omega B)\text{,}\) and the only differential that can hit it is
\begin{equation*} d^2 \colon E^2_{2,0} \;=\; H_2(B) \;\longrightarrow\; E^2_{0,1} \;=\; H_1(\Omega B). \end{equation*}
For this to leave \(0\) at \(E^\infty\) on both sides of the differential, \(d^2\) must be an isomorphism. We conclude:
\begin{equation*} H_1(\Omega B) \;\cong\; H_2(B). \end{equation*}
This is a baby version of the kind of differential-finding argument that drives essentially every Serre spectral sequence computation.

Example 18.10. Cohomological Serre for \(S^1 \to S^\infty \to \mathbb{C}P^\infty\).

The Hopf fibration stabilises to a fibration \(S^1 \to S^\infty \to \mathbb{C}P^\infty\) with contractible total space \(S^\infty \simeq *\text{.}\) The cohomological Serre spectral sequence is
\begin{equation*} E_2^{p,q} \;=\; H^p\bigl(\mathbb{C}P^\infty;\, H^q(S^1)\bigr) \;\Longrightarrow\; H^{p+q}(S^\infty). \end{equation*}
Since \(H^*(\mathbb{C}P^\infty; \mathbb{Z}) = \mathbb{Z}[x]\) with \(|x| = 2\) and \(H^*(S^1) = \Lambda(\alpha)\) with \(|\alpha| = 1\text{,}\) the \(E_2\) page in the relevant region is:
Recall that \(d_2\) in the cohomological Serre grading goes \(E_2^{p,q} \to E_2^{p+2, q-1}\text{.}\) Since \(H^*(S^\infty) = 0\) in positive degrees, the \(E_\infty\) page must be zero off the origin. The only differentials that can possibly achieve this are \(d_2 \colon E_2^{0,1} \to E_2^{2,0}\text{,}\) \(d_2 \colon E_2^{2,1} \to E_2^{4,0}\text{,}\) and so on, all of which must therefore be isomorphisms. The fundamental class \(\alpha \in H^1(S^1)\) transgresses across the fibration to the generator \(x = d_2(\alpha) \in H^2(\mathbb{C}P^\infty)\text{;}\) this is a paradigmatic example of transgression.
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