Ordinary Homology Theories
These are the theories satisfying the "dimension axiom" of the Eilenberg–Steenrod axioms that the homology of a point vanishes in dimension other than 0. They are determined by an abelian coefficient group G, and denoted by H(X, G) (where G is sometimes omitted, especially if it is Z). Usually G is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p.
The cohomology functors of ordinary cohomology theories are represented by Eilenberg–MacLane spaces.
On simplicial complexes, these theories coincide with singular homology and cohomology.
Read more about this topic: List Of Cohomology Theories
Famous quotes containing the words ordinary and/or theories:
“How far men go for the material of their houses! The inhabitants of the most civilized cities, in all ages, send into far, primitive forests, beyond the bounds of their civilization, where the moose and bear and savage dwell, for their pine boards for ordinary use. And, on the other hand, the savage soon receives from cities iron arrow-points, hatchets, and guns, to point his savageness with.”
—Henry David Thoreau (1817–1862)
“The real trouble about women is that they must always go on trying to adapt themselves to men’s theories of women, as they always have done. When a woman is thoroughly herself, she is being what her type of man wants her to be. When a woman is hysterical it’s because she doesn’t quite know what to be, which pattern to follow, which man’s picture of woman to live up to.”
—D.H. (David Herbert)