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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. Homology and cohomology with integer coefficients. Spectrum: H (Eilenberg–MacLane spectrum of the integers.) Coefficient ring: πn(H) = Z if n = 0, 0 otherwise. The original homology theory. Homology and cohomology with rational (or real or complex) coefficients. Spectrum: HQ (Eilenberg–Mac Lane spectrum of the rationals.) Coefficient ring: πn(HQ) = Q if n = 0, 0 otherwise. These are the easiest of all homology theories. The homology groups HQn(X) are often denoted by Hn(X, Q). The homology groups H(X, Q), H(X, R), H(X, C) with rational, real, and complex coefficients are all similar, and are used mainly when torsion is not of interest (or too complicated to work out). The Hodge decomposition writes the complex cohomology of a complex projective variety as a sum of sheaf cohomology groups. Homology and cohomology with mod p coefficients. Spectrum: HZp (Eilenberg–Maclane spectrum of the integers mod p.) Coefficient ring: πn(HZp) = Zp (Integers mod p) if n = 0, 0 otherwise. |
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