Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects
"holes" in a space. Cohomology has more algebraic
structure than homology, making it into a graded
ring (with multiplication given by the so-called "cup
product"), whereas homology is just a graded
Abelian group invariant of a space.

A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrod
axioms with the exception of the dimension axiom.

## See also

Aleksandrov-Čech Cohomology,

Alexander-Spanier Cohomology,

Čech Cohomology,

Cup
Product,

de Rham Cohomology,

Exotic
Cohomology,

Graded Algebra,

Homology
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## References

Rabson, D. A.; Huesman, J. F.; Fisher, B. N. "Cohomology for Anyone." *Found. Phys.* **33**, 1769-1796, 2003.## Referenced
on Wolfram|Alpha

Cohomology
## Cite this as:

Weisstein, Eric W. "Cohomology." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Cohomology.html

## Subject classifications