Section 18.2 What Do We Use Them For?

Spectral sequences are primarily used to compute (co)homology and homotopy groups. They are similar in spirit to long exact sequences, but contain more information at the cost of being more elaborate.

Example 18.1. Künneth / Tor spectral sequence.

Given two chain complexes \(C_*\) and \(D_*\) over a ring \(R\text{,}\) there is a spectral sequence

\begin{equation*} E^2_{p,q} \;=\; \mathrm{Tor}^R_{p,q}\bigl(H_*(C),\, H_*(D)\bigr) \;\Longrightarrow\; H_{p+q}(C \otimes^{\mathbb{L}} D) \end{equation*}

computing the homology of the (derived) tensor product from the homologies of the factors.

Example 18.2. Serre spectral sequence.

Given a Serre fibration \(F \to E \to B\) with \(B\) simply connected, there is a spectral sequence

\begin{equation*} E^2_{p,q} \;=\; H_p\bigl(B;\, H_q(F)\bigr) \;\Longrightarrow\; H_{p+q}(E) \end{equation*}

computing the homology of the total space from that of the base and the fibre. There is also a cohomological version.

Example 18.3. Atiyah–Hirzebruch spectral sequence.

Given a generalized cohomology theory \(E^*\) and a CW complex \(X\text{,}\) there is a spectral sequence

\begin{equation*} E_2^{p,q} \;=\; H^p\bigl(X;\, E^q(\mathrm{pt})\bigr) \;\Longrightarrow\; E^{p+q}(X) \end{equation*}

computing the generalized cohomology of \(X\) from its ordinary cohomology with coefficients in the homotopy groups \(E^*(\mathrm{pt})\text{.}\)

Prev Top Next