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:
“The hearts of small children are delicate organs. A cruel beginning in this world can twist them into curious shapes. The heart of a hurt child can shrink so that forever afterward it is hard and pitted as the seed of a peach. Or, again, the heart of such a child may fester and swell until it is misery to carry within the body, easily chafed and hurt by the most ordinary things.”
—Carson McCullers (1917–1967)
“The two most far-reaching critical theories at the beginning of the latest phase of industrial society were those of Marx and Freud. Marx showed the moving powers and the conflicts in the social-historical process. Freud aimed at the critical uncovering of the inner conflicts. Both worked for the liberation of man, even though Marx’s concept was more comprehensive and less time-bound than Freud’s.”
—Erich Fromm (1900–1980)