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 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. [1]
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 .
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
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. [2]
One of the most important property of this Alexander cohomology module with compact support is the following theorem:
as . Hence if , and 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.
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 . [5] 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 [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.
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 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. [1]
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 .
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
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. [2]
One of the most important property of this Alexander cohomology module with compact support is the following theorem:
as . Hence if , and 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.
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 . [5] 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 [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.