Alexander–Spanier cohomology
# Alexander–Spanier cohomology

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

## History

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.

## Definition

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 $C^{p}$ is the set of all functions from $X^{p+1}$ to G with differential $d\colon C^{p-1}\to C^{p}$ given by

    $df(x_{0},\ldots ,x_{p})=\sum _{i}(-1)^{i}f(x_{0},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{p}).$

The defined cochain complex $C^{*}(X;G)$ does not rely on the topology of $X$. In fact, if $X$ is a nonempty space, $G\simeq H^{*}(C^{*}(X;G))$ where $G$ is a graded module whose only nontrivial module is $G$ at degree 0.

An element $\varphi \in C^{p}(X)$ is said to be locally zero if there is a covering $\{U\}$ of $X$ by open sets such that $\varphi$ vanishes on any $(p+1)$-tuple of $X$ which lies in some element of $\{U\}$ (i.e. $\varphi$ vanishes on ⋃$_{U∈{U}}$$U^{p+1}{\textstyle$ \bigcup _{U\in \{U\}}U^{p+1}}). The subset of $C^{p}(X)$ consisting of locally zero functions is a submodule, denote by $C_{0}^{p}(X)$. $C_{0}^{*}(X)=\{C_{0}^{p}(X),d\}$ is a cochain subcomplex of $C^{*}(X)$ so we define a quotient cochain complex ${\bar {C}}^{*}(X)=C^{*}(X)/C_{0}^{*}(X)$. The Alexander–Spanier cohomology groups ${\bar {H}}^{p}(X,G)$ are defined to be the cohomology groups of ${\bar {C}}^{*}(X)$.

### Induced homomorphism

Given a function $f:X\to Y$ which is not necessarily continuous, there is an induced cochain map

    $f^{\sharp }:C^{*}(Y;G)\to C^{*}(X;G)$

defined by $(f^{\sharp }\varphi )(x_{0},...,x_{p})=(\varphi f)(x_{0},...,x_{p}),\ \varphi \in C^{p}(Y);\ x_{0},...,x_{p}\in X$

If $f$ is continuous, there is an induced cochain map

    $f^{\sharp }:{\bar {C}}^{*}(Y;G)\to {\bar {C}}^{*}(X;G)$

### Relative cohomology module

If $A$ is a subspace of $X$ and $i:A\hookrightarrow X$ is an inclusion map, then there is an induced epimorphism $i^{\sharp }:{\bar {C}}^{*}(X;G)\to {\bar {C}}^{*}(A;G)$. The kernel of $i^{\sharp }$ is a cochain subcomplex of ${\bar {C}}^{*}(X;G)$ which is denoted by ${\bar {C}}^{*}(X,A;G)$. If $C^{*}(X,A)$ denote the subcomplex of $C^{*}(X)$ of functions $\varphi$ that are locally zero on $A$, then ${\bar {C}}^{*}(X,A)=C^{*}(X,A)/C_{0}^{*}(X)$.

The relative module is ${\bar {H}}^{*}(X,A;G)$ is defined to be the cohomology module of ${\bar {C}}^{*}(X,A;G)$.

${\bar {H}}^{q}(X,A;G)$ is called the Alexander cohomology module of $(X,A)$ of degree $q$ with coefficients $G$ 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 $X$ is a one-point space, $G\simeq {\bar {H}}^{*}(X;G)$
* (Exactness axiom) If $(X,A)$ is a topological pair with inclusion maps $i:A\hookrightarrow X$ and $j:X\hookrightarrow (X,A)$, there is an exact sequence $\cdots \to {\bar {H}}^{q}(X,A;G)\xrightarrow {j^{*}} {\bar {H}}^{q}(X;G)\xrightarrow {i^{*}} {\bar {H}}^{q}(A;G)\xrightarrow {\delta ^{*}} {\bar {H}}^{q+1}(X,A;G)\to \cdots$
* (Excision axiom) For topological pair $(X,A)$, if $U$ is an open subset of $X$ such that ${\bar {U}}\subset \operatorname {int} A$, then ${\bar {C}}^{*}(X,A)\simeq {\bar {C}}^{*}(X-U,A-U)$.
* (Homotopy axiom) If $f_{0},f_{1}:(X,A)\to (Y,B)$ are homotopic, then $f_{0}^{*}=f_{1}^{*}:H^{*}(Y,B;G)\to H^{*}(X,A;G)$

## Alexander cohomology with compact supports

A subset $B\subset X$ is said to be cobounded if $X-B$ 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 $(X,A)$ by adding the property that $\varphi \in C^{q}(X,A;G)$ is locally zero on some cobounded subset of $X$.

Formally, one can define as follows : For given topological pair $(X,A)$, the submodule $C_{c}^{q}(X,A;G)$ of $C^{q}(X,A;G)$ consists of $\varphi \in C^{q}(X,A;G)$ such that $\varphi$ is locally zero on some cobounded subset of $X$.

Similar to the Alexander cohomology module, one can get a cochain complex $C_{c}^{*}(X,A;G)=\{C_{c}^{q}(X,A;G),\delta \}$ and a cochain complex ${\bar {C}}_{c}^{*}(X,A;G)=C_{c}^{*}(X,A;G)/C_{0}^{*}(X;G)$.

The cohomology module induced from the cochain complex ${\bar {C}}_{c}^{*}$ is called the Alexander cohomology of $(X,A)$ with compact supports and denoted by ${\bar {H}}_{c}^{*}(X,A;G)$. 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 $\delta ^{*}:{\bar {H}}_{c}^{q}(A;G)\to {\bar {H}}_{c}^{q+1}(X,A;G)$ only when $A\subset X$ 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 $X$ is a locally compact Hausdorff space and $X^{+}$ is the one-point compactification of $X$, then there is an isomorphism ${\bar {H}}_{c}^{q}(X;G)\simeq {\tilde {\bar {H}}}^{q}(X^{+};G).$

### Example

    ${\bar {H}}_{c}^{q}(\mathbb {R} ^{n};G)\simeq {\begin{cases}0&q\neq n\\G&q=n\end{cases}}$

as $(\mathbb {R} ^{n})^{+}\cong S^{n}$. Hence if $n\neq m$, $\mathbb {R} ^{n}$ and $\mathbb {R} ^{m}$ are not of the same proper homotopy type.

## Relation with tautness

* From the fact that a closed subspace of a paracompact Hausdorff space is a taut subspace relative to the Alexander cohomology theory and the first Basic property of tautness, if $B\subset A\subset X$ where $X$ is a paracompact Hausdorff space and $A$ and $B$ are closed subspaces of $X$, then $(A,B)$ is taut pair in $X$ relative to the Alexander cohomology theory.

Using this tautness property, one can show the following two facts:

* (Strong excision property) Let $(X,A)$ and $(Y,B)$ be pairs with $X$ and $Y$ paracompact Hausdorff and $A$ and $B$ closed. Let $f:(X,A)\to (Y,B)$ be a closed continuous map such that $f$ induces a one-to-one map of $X-A$ onto $Y-B$. Then for all $q$ and all $G$, $f^{*}:{\bar {H}}^{q}(Y,B;G)\xrightarrow {\sim } {\bar {H}}^{q}(X,A;G)$
* (Weak continuity property) Let $\{(X_{\alpha },A_{\alpha })\}_{\alpha }$ be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let (X, $A)=(⋂X_α,$ $⋂A_α){\textstyle$ (X, A)=(\bigcap X_{\alpha }, \bigcap A_{\alpha })}. The inclusion maps $i_{\alpha }:(X,A)\to (X_{\alpha },A_{\alpha })$ induce an isomorphism

    $\{i_{\alpha }^{*}\}:\varinjlim {\bar {H}}^{q}(X_{\alpha },A_{\alpha };M)\xrightarrow {\sim } {\bar {H}}^{q}(X,A;M)$.

## 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 $X$ is connected if and only if $G\simeq {\bar {H}}^{0}(X;G)$. Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.

If $\{U_{j}\}$ is an open covering of $X$ by pairwise disjoint sets, then there is a natural isomorphism $H^¯^q(X;G)≃∏_jH^¯^q(U_j;G){\textstyle$ {\bar {H}}^{q}(X;G)\simeq \prod _{j}{\bar {H}}^{q}(U_{j};G)}. In particular, if $\{C_{j}\}$ is the collection of components of a locally connected space $X$, there is a natural isomorphism $H^¯^q(X;G)≃∏_jH^¯^q(C_j;G){\textstyle$ {\bar {H}}^{q}(X;G)\simeq \prod _{j}{\bar {H}}^{q}(C_{j};G)}.

### Variants

It is also possible to define Alexander–Spanier homology and Alexander–Spanier cohomology with compact supports. (Bredon 1997)

## Connection to other cohomologies

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

