---
tag: H28PSZR7XG
kind: concept
name: Generalized cohomology theory
msc: [55N20]
sources:
  - url: https://en.wikipedia.org/wiki/Cohomology
---

## Summary

A generalized (or extraordinary) cohomology theory is a contravariant functor from pairs of spaces to graded abelian groups.
It satisfies all the [[Eilenberg–Steenrod axioms|YHK51GHK2T]] except the dimension axiom.
So the cohomology of a point may be nonzero in many degrees.
Topological K-theory, complex cobordism and stable cohomotopy are the main examples, and each such theory is represented by a spectrum.

## Explanations

### L1 human

Ordinary cohomology gives a point only one nonzero group, in degree $0$.
Some useful invariants of spaces behave exactly like cohomology in every other way, but give a point a whole sequence of nonzero groups.
These are the generalized cohomology theories.
For example, complex K-theory, built from vector bundles, assigns to a point the integers in every even degree and zero in every odd degree.