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Atiyah-Hirzebruch spectral sequence
- Relates \({\mathbb{E}}^i(X)\) a cohomology theory to \(H^i_{\mathrm{sing}}\): \begin{align*} E^2_{p, q} = H^p(B; {\mathbb{E}}^qF) \Rightarrow{\mathbb{E}}^{p+q}X \end{align*}
From Arun:
Atiyah and Hirzebruch is good but quite telegraphic, and the AHSS just … isn’t discussed in McCleary’s book Maunder, “The spectral sequence of an extraordinary cohomology theory” might be one such reference; for example, it shows the first nonzero differential is a k-invariant. k-invariants are often written very confusingly; for example, for \({\operatorname{KU}}\), Atiyah-Hirzebruch write \(\operatorname{Sq}_3\), which you’re supposed to know means reduce mod 2, do \(\operatorname{Sq}^2\), then take the integral Bockstein.
