Generalized cohomology
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General Introduction
The phrase "generalized cohomology" is usually used to refer to a cohomology theory for topological spaces which satisfies the (generalized) Eilenberg-Steenrod axioms. The first such theory to appear, except for "ordinary" (singular) cohomology was topological K-theory.
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 Eilenberg-Steenrod axioms, following Kono-Tamaki. Here we consider CW-complexes 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"
<]]> - I, II, III: Functoriality of
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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
.
Mayer-Vietoris 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/k-theory-as-a-generalized-cohomology-theory
http://mathoverflow.net/questions/29424/difference-between-represented-and-singular-cohomology
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