Tags: #stable-homotopy
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Axioms: if we drop \(H^* {\operatorname{pt}}= 0\), we get generalized alternatives to cohomology.
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 pretty 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 Topological modular formswhich 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 operations 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.
Unsorted
Flipping roles, classifying spaces being the canonical examples being the canonical examples).
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 References. Each generalized cohomology theory determines some amount of formal group data.
Via things like BP-theory and the Ravenel conjectures.
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.
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*}