In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
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.
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
given by The defined cochain complex
consisting of locally zero functions is a submodule, denote by
so we define a quotient cochain complex
are defined to be the cohomology groups of
which is not necessarily continuous, there is an induced cochain map defined by
is continuous, there is an induced cochain map If
is an inclusion map, then there is an induced epimorphism
is defined to be the cohomology module of
is called the Alexander cohomology module of
and this module satisfies all cohomology axioms.
The resulting cohomology theory is called the Alexander (or Alexander-Spanier) cohomology theory A subset
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
Formally, one can define as follows : For given topological pair
Similar to the Alexander cohomology module, one can get a cochain complex
The cohomology module induced from the cochain complex
is called the Alexander cohomology of
Under this definition, we can modify homotopy axiom for cohomology to a proper homotopy axiom if we define a coboundary homomorphism
Similarly, excision axiom can be modified to proper excision axiom i.e. the excision map is a proper map.
[2] One of the most important property of this Alexander cohomology module with compact support is the following theorem: as
are not of the same proper homotopy type.
Using this tautness property, one can show the following two facts:[4] Recall that the singular cohomology module of a space is the direct product of the singular cohomology modules of its path components.
Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.
by pairwise disjoint sets, then there is a natural isomorphism
is the collection of components of a locally connected space
It is also possible to define Alexander–Spanier homology[6] and Alexander–Spanier cohomology with compact supports.
(Bredon 1997) The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.