In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for  topological spaces.

History
It was introduced by  for the special case of compact metric spaces, and by  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  is the set of all functions from  to G with differential  given by

The defined cochain complex  does not rely on the topology of . In fact, if  is a nonempty space,  where  is a graded module whose only nontrivial module is  at degree 0.

An element  is said to be locally zero if there is a covering  of  by open sets such that  vanishes on any -tuple of  which lies in some element of  (i.e.  vanishes on ).
The subset of  consisting of locally zero functions is a submodule, denote by .
 is a cochain subcomplex of  so we define a quotient cochain complex .
The Alexander–Spanier cohomology groups  are defined to be the cohomology groups of .

Induced homomorphism
Given a function  which is not necessarily continuous, there is an induced cochain map

defined by 

If  is continuous, there is an induced cochain map

Relative cohomology module
If  is a subspace of  and  is an inclusion map, then there is an induced epimorphism . The kernel of  is a cochain subcomplex of  which is denoted by . If  denote the subcomplex of  of functions  that are locally zero on , then .

The relative module is   is defined to be the cohomology module of .

 is called the Alexander cohomology module of  of degree  with coefficients  and this module satisfies all cohomology axioms. The resulting cohomology theory is called the Alexander (or Alexander-Spanier) cohomology theory

Cohomology theory axioms
 (Dimension axiom) If  is a one-point space, 
 (Exactness axiom) If  is a topological pair with inclusion maps  and , there is an exact sequence 
 (Excision axiom) For topological pair , if  is an open subset of  such that , then .
 (Homotopy axiom) If  are homotopic, then

Alexander cohomology with compact supports
A subset  is said to be cobounded if  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  by adding the property that  is locally zero on some cobounded subset of .

Formally, one can define as follows : For given topological pair , the submodule  of  consists of  such that  is locally zero on some cobounded subset of .

Similar to the Alexander cohomology module, one can get a cochain complex  and a cochain complex .

The cohomology module induced from the cochain complex  is called the Alexander cohomology of  with compact supports and denoted by . 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  only when  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:
 If  is a locally compact Hausdorff space and  is the one-point compactification of , then there is an isomorphism

Example

as . Hence if ,  and  are not of the same proper homotopy type.

Relation with tautness
From the fact that a closed subspace of a paracompact Hausdorff space is a taut subspace relative to the Alexander cohomology theory and the first Basic property of tautness, if  where  is a paracompact Hausdorff space and  and  are closed subspaces of , then  is taut pair in  relative to the Alexander cohomology theory.

Using this tautness property, one can show the following two facts:
 (Strong excision property) Let  and  be pairs with  and  paracompact Hausdorff and  and  closed. Let  be a closed continuous map such that  induces a one-to-one map of  onto . Then for all  and all , 
 (Weak continuity property) Let  be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let . The inclusion maps  induce an isomorphism
.

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  is connected if and only if . Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.

If  is an open covering of  by pairwise disjoint sets, then there is a natural isomorphism . In particular, if  is the collection of components of a locally connected space , there is a natural isomorphism .

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

Bibliography

Cohomology theories
Duality theories