Топологиялық кеңістіктер үшін когомология теориясы

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

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.

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

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.

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

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 .

It is also possible to define Alexander–Spanier homology and Alexander–Spanier cohomology with compact supports.

The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.