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,
Dolbeault
Cohomology,
Equivariant Cohomology,
Exotic Cohomology,
Graded
Algebra,
Group Cohomology,
Homology
Explore with Wolfram|Alpha
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
