Cohomology is an invariant of a topological space , formally "dual" to homology , and
so it detects "holes" in a space .
With coefficients in a commutative ring , singular
cohomology is a graded ring with multiplication
given by the cup product , whereas homology
gives graded Abelian groups .
A generalized homology or cohomology theory satisfies the Eilenberg-Steenrod axioms without
requiring the dimension axiom.
See also Aleksandrov-Čech Cohomology ,
Alexander-Spanier Cohomology ,
Čech Cohomology ,
Cup
Product ,
de Rham Cohomology ,
Exotic
Cohomology ,
Galois Cohomology ,
Graded
Algebra ,
Homology ,
Singular
Cohomology
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References Hatcher, A. Algebraic Topology. Rabson, D. A.; Huesman, J. F.; and 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 https://mathworld.wolfram.com/Cohomology.html
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