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

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

The defined cochain complex ${\displaystyle C^{*}(X;G)}$  does not rely on the topology of ${\displaystyle X}$ . In fact, if ${\displaystyle X}$  is a nonempty space, ${\displaystyle G\simeq H^{*}(C^{*}(X;G))}$  where ${\displaystyle G}$  is a graded module whose only nontrivial module is ${\displaystyle G}$  at degree 0.[1]

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

### Induced homomorphism

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

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

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

If ${\displaystyle f}$  is continuous, there is an induced cochain map

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

### relative cohomology module

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

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

${\displaystyle {\bar {H}}^{q}(X,A;G)}$  is called the Alexander cohomology module of ${\displaystyle (X,A)}$  of degree ${\displaystyle q}$  with coefficients ${\displaystyle G}$  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 ${\displaystyle X}$  is a one-point space, ${\displaystyle G\simeq {\bar {H}}^{*}(X;G)}$
• (Exactness axiom) If ${\displaystyle (X,A)}$  is a topological pair with inclusion maps ${\displaystyle i:A\hookrightarrow X}$  and ${\displaystyle j:X\hookrightarrow (X,A)}$ , there is an exact sequence
${\displaystyle \cdots \to {\bar {H}}^{q}(X,A;G)\xrightarrow {j^{*}} {\bar {H}}^{q}(X;G)\xrightarrow {i^{*}} {\bar {H}}^{q}(A;G)\xrightarrow {\delta ^{*}} {\bar {H}}^{q+1}(X,A;G)\to \cdots }$

• (Excision axiom) For topological pair ${\displaystyle (X,A)}$ , if ${\displaystyle U}$  is an open subset of ${\displaystyle X}$  such that ${\displaystyle {\bar {U}}\subset \operatorname {int} A}$ , then ${\displaystyle {\bar {C}}^{*}(X,A)\simeq {\bar {C}}^{*}(X-U,A-U)}$ .
• (Homotopy axiom) If ${\displaystyle f_{0},f_{1}:(X,A)\to (Y,B)}$  are homotopic, then ${\displaystyle f_{0}^{*}=f_{1}^{*}:H^{*}(Y,B;G)\to H^{*}(X,A;G)}$

## Alexander cohomology with compact supports

A subset ${\displaystyle B\subset X}$  is said to be cobounded if ${\displaystyle X-B}$  is bounded.

Similar to the definition of Alexander cohomology module, one can define Alexander cohomology module with compact supports of a pair ${\displaystyle (X,A)}$  by adding the property that ${\displaystyle \varphi \in C^{q}(X,A;G)}$  is locally zero on some cobounded subset of ${\displaystyle X}$ .

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

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

The cohomology module induced from the cochain complex ${\displaystyle {\bar {C}}_{c}^{*}}$  is called the Alexander cohomology of ${\displaystyle (X,A)}$  with compact supports and denoted by ${\displaystyle {\bar {H}}_{c}^{*}(X,A;G)}$ . 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 ${\displaystyle \delta ^{*}:{\bar {H}}_{c}^{q}(A;G)\to {\bar {H}}_{c}^{q+1}(X,A;G)}$  only when ${\displaystyle A\subset X}$  is a closed subset. Similarly, excision axiom can be modified to proper excision axiom i.e. the excision map is a proper map.[2]

### Property

One of the most important property of this Alexander cohomology module with compact support is the following theorem

• If ${\displaystyle X}$  is locally compact Hausdorff space and ${\displaystyle X^{+}}$  is the one-point compactification of ${\displaystyle X}$ , there is an isomorphism
${\displaystyle {\bar {H}}_{c}^{q}(X;G)\simeq {\tilde {\bar {H}}}^{q}(X^{+};G)}$

### Example

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

as ${\displaystyle (\mathbb {R} ^{n})^{+}\cong S^{n}}$ . Hence if ${\displaystyle n\neq m}$ , ${\displaystyle \mathbb {R} ^{n}}$  and ${\displaystyle \mathbb {R} ^{m}}$  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 [3] and the first Basic property of tautness, if ${\displaystyle B\subset A\subset X}$  where ${\displaystyle X}$  is a paracompact Hausdorff space and ${\displaystyle A}$  and ${\displaystyle B}$  are closed subspaces of ${\displaystyle X}$ , then ${\displaystyle (A,B)}$  is taut pair in ${\displaystyle X}$  relative to the Alexander cohomology theory.

Using this tautness property, one can show the following two facts[4]

• (Strong excision property) Let ${\displaystyle (X,A)}$  and ${\displaystyle (Y,B)}$  be pairs with ${\displaystyle X}$  and ${\displaystyle Y}$  paracompact Hausdorff and ${\displaystyle A}$  and ${\displaystyle B}$  closed. Let ${\displaystyle f:(X,A)\to (Y,B)}$  be a closed continuous map such that ${\displaystyle f}$  induces a one-to-one map of ${\displaystyle X-A}$  onto ${\displaystyle Y-B}$ . Then for all ${\displaystyle q}$  and all ${\displaystyle G}$ ,
${\displaystyle f^{*}:{\bar {H}}^{q}(Y,B;G)\xrightarrow {\sim } {\bar {H}}^{q}(X,A;G)}$

• (Weak continuity property) Let ${\displaystyle \{(X_{\alpha },A_{\alpha })\}_{\alpha }}$  be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let ${\textstyle (X,A)=(\bigcap X_{\alpha },\bigcap A_{\alpha })}$ . The inclusion maps ${\displaystyle i_{\alpha }:(X,A)\to (X_{\alpha },A_{\alpha })}$  induce an isomoprhim
${\displaystyle \{i_{\alpha }^{*}\}:\varinjlim {\bar {H}}^{q}(X_{\alpha },A_{\alpha };M)\xrightarrow {\sim } {\bar {H}}^{q}(X,A;M)}$

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

If ${\displaystyle \{U_{j}\}}$  is an open covering of ${\displaystyle X}$  by pairwise disjoint sets, then there is a natural isomorphism ${\textstyle {\bar {H}}^{q}(X;G)\simeq \prod _{j}{\bar {H}}^{q}(U_{j};G)}$ .[5] In particular, if ${\displaystyle \{C_{j}\}}$  is the collection of components of a locally connected space ${\displaystyle X}$ , there is a natural isomorphism ${\textstyle {\bar {H}}^{q}(X;G)\simeq \prod _{j}{\bar {H}}^{q}(C_{j};G)}$

### Variants

It is also possible to define Alexander–Spanier homology,.[6]

## 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.

## References

1. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 307. ISBN 978-0387944265.
2. ^ Spanier, Edwin H. (1966). Algebraic topology. pp. 320, 322. ISBN 978-0387944265.
3. ^ Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American Mathematical Society. 52: 441–442.
4. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 318. ISBN 978-0387944265.
5. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 310. ISBN 978-0387944265.
6. ^