---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "Atiyah Hirzebruch spectral sequence"
aliases: ["Atiyah-Hirzebruch spectral sequence", AHSS]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #spectral-sequences 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# Atiyah-Hirzebruch spectral sequence

- Relates $\EE^i(X)$ a cohomology theory to $H^i_\sing$:
$$
E^2_{p, q} = H^p(B; \EE^qF) \abuts \EE^{p+q}X
$$

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-invariant).
> k-invariants are often written very confusingly; for example, for $\KU$, Atiyah-Hirzebruch write $\Sq_3$, which you're supposed to know means reduce mod 2, do $\Sq^2$, then take the integral [Bockstein](Bockstein.md).

![](attachments/Pasted%20image%2020220403190953.png)