Cohomology theory for topological spaces
In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
History.
It was introduced by James W. Alexander (1935) for the special case of compact metric spaces, and by Edwin H. Spanier (1948) for all topological spaces, based on a suggestion of Alexander D. Wallace.
Definition.
If "X" is a topological space and "G" is an "R" module where "R" is a ring with unity, then there is a cochain complex "C" whose "p"-th term formula_1 is the set of all functions from formula_2 to "G" with differential formula_3 given by
formula_4
The defined cochain complex formula_5 does not rely on the topology of formula_6. In fact, if formula_6 is a nonempty space, formula_8 where formula_9 is a graded module whose only nontrivial module is formula_9 at degree 0.
An element formula_11 is said to be "locally zero" if there is a covering formula_12 of formula_6 by open sets such that formula_14 vanishes on any formula_15-tuple of formula_6 which lies in some element of formula_12 (i.e. formula_14 vanishes on formula_19).
The subset of formula_20 consisting of locally zero functions is a submodule, denote by formula_21.
formula_22 is a cochain subcomplex of formula_23 so we define a quotient cochain complex formula_24.
The Alexander–Spanier cohomology groups formula_25 are defined to be the cohomology groups of formula_26.
Induced homomorphism.
Given a function formula_27 which is not necessarily continuous, there is an induced cochain map
formula_28
defined by formula_29
If formula_30 is continuous, there is an induced cochain map
formula_31
Relative cohomology module.
If formula_32 is a subspace of formula_6 and formula_34 is an inclusion map, then there is an induced epimorphism formula_35. The kernel of formula_36 is a cochain subcomplex of formula_37 which is denoted by formula_38. If formula_39 denote the subcomplex of formula_23 of functions formula_14 that are locally zero on formula_32, then formula_43.
The "relative module" is formula_44 is defined to be the cohomology module of formula_38.
formula_46 is called the "Alexander cohomology module of formula_47 of degree formula_48 with coefficients formula_9" and this module satisfies all cohomology axioms. The resulting cohomology theory is called the "Alexander (or Alexander-Spanier) cohomology theory"
Alexander cohomology with compact supports.
A subset formula_63 is said to be "cobounded" if formula_64 is bounded, i.e. its closure is compact.
Similar to the definition of Alexander cohomology module, one can define Alexander cohomology module with "compact supports" of a pair formula_47 by adding the property that formula_66 is locally zero on some cobounded subset of formula_6.
Formally, one can define as follows : For given topological pair formula_47, the submodule formula_69 of formula_70 consists of formula_66 such that formula_14 is locally zero on some cobounded subset of formula_6.
Similar to the Alexander cohomology module, one can get a cochain complex formula_74 and a cochain complex formula_75.
The cohomology module induced from the cochain complex formula_76 is called the "Alexander cohomology of formula_47 with compact supports" and denoted by formula_78. Induced homomorphism of this cohomology is defined as the Alexander cohomology theory.
Under this definition, we can modify "homotopy axiom" for cohomology to a "proper homotopy axiom" if we define a coboundary homomorphism formula_79 only when formula_80 is a "closed" subset. Similarly, "excision axiom" can be modified to "proper excision axiom" i.e. the excision map is a proper map.
Property.
One of the most important property of this Alexander cohomology module with compact support is the following theorem:
formula_85
Example.
as formula_86. Hence if formula_87, formula_88 and formula_89 are not of the same "proper" homotopy type.
Relation with tautness.
Using this tautness property, one can show the following two facts:
Difference from singular cohomology theory.
Recall that the singular cohomology module of a space is the direct product of the singular cohomology modules of its path components.
A nonempty space formula_6 is connected if and only if formula_115. Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.
If formula_116 is an open covering of formula_6 by pairwise disjoint sets, then there is a natural isomorphism formula_118. In particular, if formula_119 is the collection of components of a locally connected space formula_6, there is a natural isomorphism formula_121.
Variants.
It is also possible to define Alexander–Spanier homology and Alexander–Spanier cohomology with compact supports. 
Connection to other cohomologies.
The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.
References.
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