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
![{\displaystyle df(x_{0},\ldots ,x_{p})=\sum _{i}(-1)^{i}f(x_{0},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{p}).}](//wikimedia.org/api/rest_v1/media/math/render/svg/16a135662d366f1f0a25a7d239167aff9380a866)
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
.
Induced homomorphism
Given a function
which is not necessarily continuous, there is an induced cochain map
![{\displaystyle f^{\sharp }:C^{*}(Y;G)\to C^{*}(X;G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d80c3c49a3eada176024c7fd00d39fedff65c534)
defined by ![{\displaystyle (f^{\sharp }\varphi )(x_{0},...,x_{p})=(\varphi f)(x_{0},...,x_{p}),\ \varphi \in C^{p}(Y);\ x_{0},...,x_{p}\in X}](//wikimedia.org/api/rest_v1/media/math/render/svg/95d9401ca1ccd122a252f8dd92895e3252efad6f)
If
is continuous, there is an induced cochain map
![{\displaystyle f^{\sharp }:{\bar {C}}^{*}(Y;G)\to {\bar {C}}^{*}(X;G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/6ce059619215492853a4ba08f2a07792c9b2b9f4)
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
- (Dimension axiom) If
is a one-point space, ![{\displaystyle G\simeq {\bar {H}}^{*}(X;G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/ba1ad3cc939e89b7a28554faf09d7604c0104814)
- (Exactness axiom) If
is a topological pair with inclusion maps
and
, there is an exact sequence ![{\displaystyle \cdots \to {\bar {H}}^{q}(X,A;G)\xrightarrow {j^{*}} {\bar {H}}^{q}(X;G)\xrightarrow {i^{*}} {\bar {H}}^{q}(A;G)\xrightarrow {\delta ^{*}} {\bar {H}}^{q+1}(X,A;G)\to \cdots }](//wikimedia.org/api/rest_v1/media/math/render/svg/11bb860e7269a28ec20ed9799183dcf62bdb98d7)
- (Excision axiom) For topological pair
, if
is an open subset of
such that
, then
.
- (Homotopy axiom) If
are homotopic, then ![{\displaystyle f_{0}^{*}=f_{1}^{*}:H^{*}(Y,B;G)\to H^{*}(X,A;G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/dd612e690344ee144e98886a44b021af1d578ffe)
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]
- From the fact that a closed subspace of a paracompact Hausdorff space is a taut subspace relative to the Alexander cohomology theory[3] 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:[4]
- (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
, ![{\displaystyle f^{*}:{\bar {H}}^{q}(Y,B;G)\xrightarrow {\sim } {\bar {H}}^{q}(X,A;G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/a8b44056a8ee5ac07fb9d8e4b9b9175dad36b847)
- (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
.
The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.
Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American Mathematical Society. 52: 441–442.
- Alexander, James W. (1935), "On the Chains of a Complex and Their Duals", Proceedings of the National Academy of Sciences of the United States of America, 21 (8), National Academy of Sciences: 509–511, Bibcode:1935PNAS...21..509A, doi:10.1073/pnas.21.8.509, ISSN 0027-8424, JSTOR 86360, PMC 1076641, PMID 16577676
- Bredon, Glen E. (1997), Sheaf theory, Graduate Texts in Mathematics, vol. 170 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0647-7, ISBN 978-0-387-94905-5, MR 1481706
- Massey, William S. (1978), "How to give an exposition of the Čech-Alexander-Spanier type homology theory", The American Mathematical Monthly, 85 (2): 75–83, doi:10.2307/2321782, ISSN 0002-9890, JSTOR 2321782, MR 0488017
- Massey, William S. (1978), Homology and cohomology theory. An approach based on Alexander-Spanier cochains., Monographs and Textbooks in Pure and Applied Mathematics, vol. 46, New York: Marcel Dekker Inc., ISBN 978-0-8247-6662-7, MR 0488016
- Spanier, Edwin H. (1948), "Cohomology theory for general spaces", Annals of Mathematics, Second Series, 49 (2): 407–427, doi:10.2307/1969289, ISSN 0003-486X, JSTOR 1969289, MR 0024621
- Spanier, Edwin H. (1966), Algebraic topology, ISBN 978-0387944265