Alexander–Spanier cohomology

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

History

It was introduced by Template:Harvs for the special case of compact metric spaces, and by Template:Harvs 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 $d:C^{p-1}\to C^p$ given by

$df(x_0,\dots ,x_p)=\sum _i(-1{)}^{i}f(x_0,\dots ,x_{i-1},x_{i+1},\dots ,x_p).$

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

An element $\phi \in C^p(X)$ is said to be locally zero if there is a covering $\{U\}$ of $X$ by open sets such that $\phi$ vanishes on any $(p+1)$-tuple of $X$ which lies in some element of $\{U\}$ (i.e. $\phi$ vanishes on ${\bigcup }_{U\in \{U\}}{U}^{p+1}$). The subset of $C^p(X)$ consisting of locally zero functions is a submodule, denote by $C_0^p(X)$. $C_0^{\ast }(X)=\{C_0^p(X),d\}$ is a cochain subcomplex of $C^{\ast }(X)$ so we define a quotient cochain complex ${\overline{C}}^{\ast }(X)=C^{\ast }(X)/C_0^{\ast }(X)$. The Alexander–Spanier cohomology groups ${\overline{H}}^p(X,G)$ are defined to be the cohomology groups of ${\overline{C}}^{\ast }(X)$.

Induced homomorphism

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

$f^{\mathrm{♯}}:C^{\ast }(Y;G)\to C^{\ast }(X;G)$

defined by $(f^{\mathrm{♯}}\phi )(x_0,\mathrm{...},x_p)=(\phi f)(x_0,\mathrm{...},x_p),\phi \in C^p(Y);x_0,\mathrm{...},x_p\in X$

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

$f^{\mathrm{♯}}:{\overline{C}}^{\ast }(Y;G)\to {\overline{C}}^{\ast }(X;G)$

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 $i^{\mathrm{♯}}:{\overline{C}}^{\ast }(X;G)\to {\overline{C}}^{\ast }(A;G)$. The kernel of $i^{\mathrm{♯}}$ is a cochain subcomplex of ${\overline{C}}^{\ast }(X;G)$ which is denoted by ${\overline{C}}^{\ast }(X,A;G)$. If $C^{\ast }(X,A)$ denote the subcomplex of $C^{\ast }(X)$ of functions $\phi$ that are locally zero on $A$, then ${\overline{C}}^{\ast }(X,A)=C^{\ast }(X,A)/C_0^{\ast }(X)$.

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

${\overline{H}}^q(X,A;G)$ 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, $G\simeq {\overline{H}}^{\ast }(X;G)$
• (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 $$\cdots \to {\overline{H}}^q(X,A;G)\stackrel{j^{\ast }}{\to }{\overline{H}}^q(X;G)\stackrel{i^{\ast }}{\to }{\overline{H}}^q(A;G)\stackrel{{\delta }^{\ast }}{\to }{\overline{H}}^{q+1}(X,A;G)\to \cdots$$
• (Excision axiom) For topological pair $(X,A)$, if $U$ is an open subset of $X$ such that $\overline{U}\subset \mathrm{int}A$, then ${\overline{C}}^{\ast }(X,A)\simeq {\overline{C}}^{\ast }(X-U,A-U)$.
• (Homotopy axiom) If $f_0,f_1:(X,A)\to (Y,B)$ are homotopic, then $f_0^{\ast }=f_1^{\ast }:H^{\ast }(Y,B;G)\to H^{\ast }(X,A;G)$

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 $\phi \in C^q(X,A;G)$ 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 $\phi \in C^q(X,A;G)$ such that $\phi$ is locally zero on some cobounded subset of $X$.

Similar to the Alexander cohomology module, one can get a cochain complex $C_c^{\ast }(X,A;G)=\{C_c^q(X,A;G),\delta \}$ and a cochain complex ${\overline{C}}_c^{\ast }(X,A;G)=C_c^{\ast }(X,A;G)/C_0^{\ast }(X;G)$.

The cohomology module induced from the cochain complex ${\overline{C}}_c^{\ast }$ is called the Alexander cohomology of $(X,A)$ with compact supports and denoted by ${\overline{H}}_c^{\ast }(X,A;G)$. 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 ${\delta }^{\ast }:{\overline{H}}_c^q(A;G)\to {\overline{H}}_c^{q+1}(X,A;G)$ 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 $${\overline{H}}_c^q(X;G)\simeq {\tilde{\overline{H}}}^q(X^{+};G).$$

Example

${\overline{H}}_c^q({ℝ}^n;G)\simeq \{\begin{array}{cc}0& q\ne n\\ G& q=n\end{array}$

as $({ℝ}^n{)}^{+}\cong S^n$. Hence if $n\ne m$, ${ℝ}^n$ and ${ℝ}^m$ 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 $G$, $$f^{\ast }:{\overline{H}}^q(Y,B;G)\stackrel{\sim }{\to }{\overline{H}}^q(X,A;G)$$
• (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 $(X,A)=(\bigcap X_{\alpha },\bigcap A_{\alpha })$. The inclusion maps $i_{\alpha }:(X,A)\to (X_{\alpha },A_{\alpha })$ induce an isomorphism

  $\{i_{\alpha }^{\ast }\}:\underrightarrow{\lim }{\overline{H}}^q(X_{\alpha },A_{\alpha };M)\stackrel{\sim }{\to }{\overline{H}}^q(X,A;M)$.

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 $G\simeq {\overline{H}}^0(X;G)$. 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 ${\overline{H}}^q(X;G)\simeq {\prod }_j{\overline{H}}^q(U_j;G)$. In particular, if $\{C_j\}$ is the collection of components of a locally connected space $X$, there is a natural isomorphism ${\overline{H}}^q(X;G)\simeq {\prod }_j{\overline{H}}^q(C_j;G)$.

Variants

It is also possible to define Alexander–Spanier homologyTemplate:Sfn and Alexander–Spanier cohomology with compact supports. Template:Harv

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.

Bibliography

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