---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "complex oriented cohomology theory"
aliases: [complex oriented cohomology theory]
---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [FGL](Unsorted/Formal%20group.md)
	- [Landweber exactness](Unsorted/Landweber%20exactness.md)

---

# complex oriented cohomology theory

- A condition on a [generalized cohomolology theories](Unsorted/cohomolology%20theory.md) involving the [Thom class](Unsorted/Thom%20space.md):
  
![](attachments/Pasted%20image%2020220508195540.png)
![attachments/Pasted image 20210511201541.png](attachments/Pasted%20image%2020210511201541.png)

Think of this as a factorization of the [counit](counit)

\begin{tikzcd}
	\SS && {X \da \Sigma^{\infty-2}\CP^{\infty}} && E
	\arrow["{x^E}", from=1-3, to=1-5]
	\arrow["{\eta_E}"', curve={height=30pt}, from=1-1, to=1-5]
	\arrow["{\eta_X}", from=1-1, to=1-3]
\end{tikzcd}

> [https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXFNTIl0sWzIsMCwiWCBcXGRhIFxcU2lnbWFee1xcaW5mdHktMn1cXENQXntcXGluZnR5fSJdLFs0LDAsIkUiXSxbMSwyLCJ4XkUiXSxbMCwyLCJcXGV0YV9FIiwyLHsiY3VydmUiOjV9XSxbMCwxLCJcXGV0YV9YIl1d](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXFNTIl0sWzIsMCwiWCBcXGRhIFxcU2lnbWFee1xcaW5mdHktMn1cXENQXntcXGluZnR5fSJdLFs0LDAsIkUiXSxbMSwyLCJ4XkUiXSxbMCwyLCJcXGV0YV9FIiwyLHsiY3VydmUiOjV9XSxbMCwxLCJcXGV0YV9YIl1d)

A ring spectrum $E$ is complex orientable iff the [Atiyah Hirzebruch spectral sequence](Atiyah%20Hirzebruch%20spectral%20sequence.md) collapses at $E_2$:
$$
E_{2}^{p, q}=H^{p}\left(\mathbb{C} P^{\infty} ; \pi_{q}(E)\right) \Longrightarrow E^{p+q}\left(\mathbb{C} P^{\infty}\right)
$$

# Motivations

![](attachments/Pasted%20image%2020220508195645.png)
![](attachments/Pasted%20image%2020220508195655.png)
![](attachments/Pasted%20image%2020220508195926.png)