Alexander–Spanier cohomology

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 ${\displaystyle d:C^{p-1}\to C^p}$ given by

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

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

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

Induced homomorphism

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

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

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

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

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

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

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

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

Alexander cohomology with compact supports

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

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

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

Property

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

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

Example

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

as ${\displaystyle ({\mathbb{R}}^n{)}^{+}\cong S^n}$. Hence if ${\displaystyle n\ne m}$, ${\displaystyle {\mathbb{R}}^n}$ 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 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:

• (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 ${\displaystyle G}$, $${\displaystyle f^{\ast }:{\overline{H}}^q(Y,B;G)\stackrel{\sim }{\to }{\overline{H}}^q(X,A;G)}$$
• (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 $(X,A)=(\bigcap X_{\alpha },\bigcap A_{\alpha })$. The inclusion maps ${\displaystyle i_{\alpha }:(X,A)\to (X_{\alpha },A_{\alpha })}$ induce an isomorphism

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

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

If ${\displaystyle \{U_j\}}$ is an open covering of ${\displaystyle X}$ 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 ${\displaystyle \{C_j\}}$ is the collection of components of a locally connected space ${\displaystyle X}$, 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 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.