Reduced cohomology

General Introduction
See also Generalized cohomology
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Paper References
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Definition
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Properties
See Generalized cohomology for some notation.
A reduced cohomology theory is a sequence of functors together with natural transformations , satisfying the following axioms:
 IV' (Exactness): For any inclusion in , the sequence: is exact
 V' (Homotopy): Homotopic morphisms in induce the same map on cohomology
 VI' (Suspension): For , the map is an isomorphism for all .
We can also consider an additivity axiom, saying that "the cohomology of a wedge product is the product of the individual cohomology groups".
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Standard theorems
The reduced cohomology theories are in bijection with the generalized cohomology theories under composition with the functor from CW pairs to sending to .
If a map of pointed CW complexes induces an isomorphism on ordinary homology, then it also induces an isomorphism on any reduced cohomology theory. If the complexes involved are finite, the same holds for ordinary cohomology.
(Milnor exact sequence): For a theory satisfying additivity, we have, for any nested union , an exact sequence
Roughly, the following is true: A cohomology theory on finite CW complexes is determined by its Coefficient group, and the same is true for a theory on CW complexes, provided it satisfies additivity.
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
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Other Information
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