Generalized cohomology

General Introduction
The phrase "generalized cohomology" is usually used to refer to a cohomology theory for topological spaces which satisfies the (generalized) EilenbergSteenrod axioms. The first such theory to appear, except for "ordinary" (singular) cohomology was topological Ktheory.
See also Reduced cohomology
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Search results
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Online References
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Paper References
Kono and Tamaki: Generalized cohomology.
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Definition
A generalized cohomology theory is a sequence of functors satisfying axioms I to VI below.
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Properties
We present here the EilenbergSteenrod axioms, following KonoTamaki. Here we consider CWcomplexes only, it would probably be better to consider compactly generated spaces.
We consider the category of CW pairs, and the category of finite CW pairs. A CW pair is a pair consisting of a CW complex and a subcomplex , which can be empty.
There is a covariant endofunctor on sending to .
We consider a sequence of contravariant functors together with natural transformations for .
Axioms:
 I, II, III: Functoriality of and naturality of
IV (Exactness): For any , the following sequence is exact:
V (Homotopy): Homotopic maps in give same map on cohomology
 VI (Excision): Let . The inclusion induces an isomorphism on each cohomology group.
 VII (Dimension): for all nonzero .
 VIII (Additivity): "The cohomology of a disjoint union of spaces is the product of the cohomology of each space"
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Standard theorems
Any cohomology theory satisfying axioms I  VI above has the following three exact sequences.
We say that is a triple if and are CW pairs. We say that is a triad if and are CW pairs.
Exact sequence for triple:
Exact sequence for a triad:
where is the excision isomorphism followed by .
MayerVietoris exact sequence for a triad:
where is the difference map, and where , where the middle map is excision.
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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
http://mathoverflow.net/questions/18513/ktheoryasageneralizedcohomologytheory
http://mathoverflow.net/questions/29424/differencebetweenrepresentedandsingularcohomology
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