# Generalized cohomology

• ## 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.

<]]>

<]]>

<]]>
• ## Paper References

Kono and Tamaki: Generalized cohomology.

<]]>
• ## Definition

A generalized cohomology theory is a sequence of functors satisfying axioms I to VI below.

<]]>
• ## 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"
<]]>
• ## 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:  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.

<]]>

<]]>

<]]>

<]]>

<]]>

<]]>

<]]>