In
mathematics, particularly in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, Alexander–Spanier cohomology is a
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
theory for
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s.
History
It was introduced by for the special case of compact
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, 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
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
where ''R'' is a ring with unity, then there is a
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
''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
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
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 In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
of
, then there is an isomorphism
Example
:
as
. Hence if
,
and
are not of the same ''proper''
homotopy type
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
.
Relation with tautness
*From the fact that a closed subspace of a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
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
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectednes ...
, 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
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
Motivation
Let ''X'' be a topol ...
groups for compact
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s, and coincide with
singular cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
groups for locally finite complexes.
References
Bibliography
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*
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{{DEFAULTSORT:Alexander-Spanier cohomology
Cohomology theories
Duality theories