---
date: 2022-02-23 18:45
modification date: Friday 1st April 2022 21:54:46
title: "cohomolology theory"
aliases: [extraordinary cohomolology theory, extraordinary cohomolology theories, generalized cohomolology theory, generalized cohomolology theories, cohomolology theory, cohomolology theories]

---

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


---

- Tags 
	- #homotopy/stable-homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [complex oriented cohomology theory](Unsorted/complex%20oriented%20cohomology%20theory.md)
	- [Representing cohomology](Representing%20cohomology.md)
	- The main examples:
		- [topological K theory](Unsorted/topological%20K%20theory.md)
		- [bordism](Unsorted/cobordism.md)
	- [spectra](Unsorted/spectra.md)
	- [Representing cohomology](Unsorted/Representing%20cohomology.md)

---

# cohomolology theory


- When does a cohomology theory admit a ring structure? Or a [DGA](differential%20graded%20algebras)? #todo/questions


- Cohomology theories: Write $\Map(X,Y)$ for the mapping spectrum defined by the adjunction $[Z,\Map(X,Y)] = [Z\smashprod X,Y]$, then
  
$$
E^n(X) = \pi_{-n}\Maps(\Sigma^\infty X, \EE) \qquad 
E_n(X) = \pi_n(\Sigma^\infty X \smashprod \EE) = \SHC(\SS, \Sigma^\infty X \smashprod \EE)
$$


# Motivation

The basic result is [cofiber sequence](cofiber%20sequence.md) to long exact sequences and has a [suspension](suspension) isomorphism and takes wedges to products, then this is represented by a sequence of spaces $\left\{{E_n}\right\}$ with weak equivalences $E_n \cong \Omega E_{n+1}$ coming from the existence of the suspension isomorphism and the [Yoneda lemma]().

Conversely, given a sequence of spaces $\left\{{E_n}\right\}$ with maps $\Sigma E_n\to E_{n+1}$, you can cook up a cohomology theory. This means that one can do some formal manipulations inside the category of [spectra](spectra.md) and produce lots of different cohomology theories, even ones that have no geometric interpretation a priori.

For example, there is a cohomology theory called [TMF](TMF) which has (as of now) no geometric interpretation, but can detect many nontrivial maps between spheres, and can even be used to prove results in [number theory](number%20theory.md).

We see that self maps $E\to E$ of a [cohomology operation](Unsorted/cohomology%20operations.md) by cooking up maps of spectra.

There's even a machine, [Adams spectral sequence](Archive/0200_Stable%20Homotopy%20Seminar%202021/Adams%20Sseq.md), which computes all maps between spectra.

You can take the homotopy groups of a cohomology theory.

# Multiplicative

We say that a cohomology theory $E$ is **multiplicative** if its representing spectrum is endowed with a multiplication $E^{\wedge 2}\to E$ that is associative and unital up to homotopy, i.e. a [ring spectrum](Unsorted/ring%20spectra.md).

# Oriented cohomology theories

See [complex oriented cohomology theory](complex%20oriented%20cohomology%20theory.md)

# Extracting cohomology from a spectrum

- Reduced cohomology associated to a spectrum:
$$
\tilde{E}^{k}(X)=\operatorname{colim}_{n}\left[S^{\smashprod n}\smashprod X, E_{k+n}\right]
$$

# Unsorted

After a lot of hard work (with some of the bigger names including Adams, Milnor, and Quillen, though I am leaving a lot of important names out) it is discovered, starting from almost pure calculation, that the stable homotopy category has a connection to the category of 1-dimensional [formal groups](Unsorted/Formal%20group.md) via the study of [characteristic classes](Unsorted/characteristic%20class.md)


Each generalized cohomology theory determines some amount of [formal group](formal%20group.md) data via things like [BP](BP) theory and the [Ravenel conjectures](Ravenel%20conjectures).

Reduced cohomology associated to a [spectrum](Unsorted/spectra.md):
$$
\tilde{E}^{k}(X)=\operatorname{colim}_{n}\left[S^{\smashprod n}\smashprod X, E_{k+n}\right]
$$

[Freudenthal suspension](Freudenthal%20suspension) realized by a weak equivalence of [Eilenberg-MacLane spaces](Unsorted/Eilenberg-MacLane%20spaces.md):
![](attachments/Pasted%20image%2020220401215352.png)

Relation to [cohomology operations](cohomology%20operations):
![](attachments/Pasted%20image%2020220401215422.png)

Relation to [Steenrod operations](Unsorted/Steenrod%20algebra.md):
![](attachments/Pasted%20image%2020220401215445.png)