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- complex oriented cohomology theory
- Representing cohomology
- The main examples:
- spectra
- Representing cohomology
cohomolology theory
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When does a cohomology theory admit a ring structure? Or a DGA? #todo/questions
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Cohomology theories: Write \(\mathop{\mathrm{Maps}}(X,Y)\) for the mapping spectrum defined by the adjunction \([Z,\mathop{\mathrm{Maps}}(X,Y)] = [Z\wedge X,Y]\), then
\begin{align*} E^n(X) = \pi_{-n}\mathop{\mathrm{Maps}}(\Sigma^\infty X, {\mathbb{E}}) \qquad E_n(X) = \pi_n(\Sigma^\infty X \wedge{\mathbb{E}}) = {\mathsf{SHC}}({\mathbb{S}}, \Sigma^\infty X \wedge{\mathbb{E}}) \end{align*}
Motivation
The basic result is cofiber sequence to long exact sequences and has a 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 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 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.
We see that self maps \(E\to E\) of a cohomology operation by cooking up maps of spectra.
There’s even a machine, Adams spectral sequence, 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.
Oriented cohomology theories
See complex oriented cohomology theory
Extracting cohomology from a spectrum
- Reduced cohomology associated to a spectrum: \begin{align*} \tilde{E}^{k}(X)=\operatorname{colim}_{n}\left[S^{\wedge n}\wedge X, E_{k+n}\right] \end{align*}
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 via the study of characteristic classes
Each generalized cohomology theory determines some amount of formal group data via things like BP theory and the Ravenel conjectures.
Reduced cohomology associated to a spectrum: \begin{align*} \tilde{E}^{k}(X)=\operatorname{colim}_{n}\left[S^{\wedge n}\wedge X, E_{k+n}\right] \end{align*}
Freudenthal suspension realized by a weak equivalence of Eilenberg-MacLane spaces: 
Relation to cohomology operations: 
Relation to Steenrod operations: 
