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 <math>C^p</math> is the set of all functions from <math>X^{p+1}</math> to G with differential <math>d\colon C^{p-1} \to C^{p}</math> given by
- <math>df(x_0,\ldots,x_p)= \sum_i(-1)^if(x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_p).</math>
The defined cochain complex <math>C^*(X;G)</math> does not rely on the topology of <math>X</math>. In fact, if <math>X</math> is a nonempty space, <math>G\simeq H^*(C^*(X;G))</math> where <math>G</math> is a graded module whose only nontrivial module is <math>G</math> at degree 0.<ref>Template:Cite book</ref>
An element <math>\varphi\in C^p(X)</math> is said to be locally zero if there is a covering <math>\{U\}</math> of <math>X</math> by open sets such that <math>\varphi</math> vanishes on any <math>(p+1)</math>-tuple of <math>X</math> which lies in some element of <math>\{U\}</math> (i.e. <math>\varphi</math> vanishes on <math display="inline">\bigcup_{U\in\{U\}}U^{p+1}</math>). The subset of <math>C^p(X)</math> consisting of locally zero functions is a submodule, denote by <math>C_0^p(X)</math>. <math>C^*_0(X) = \{C_0^p(X),d\}</math> is a cochain subcomplex of <math>C^*(X)</math> so we define a quotient cochain complex <math>\bar{C}^*(X)=C^*(X)/C_0^*(X)</math>. The Alexander–Spanier cohomology groups <math>\bar{H}^p(X,G)</math> are defined to be the cohomology groups of <math>\bar{C}^*(X)</math>.
Induced homomorphism
Given a function <math>f:X\to Y</math> which is not necessarily continuous, there is an induced cochain map
- <math>f^\sharp:C^*(Y;G)\to C^*(X;G)</math>
defined by <math>(f^\sharp\varphi)(x_0,...,x_p) = (\varphi f)(x_0,...,x_p),\ \varphi\in C^p(Y);\ x_0,...,x_p\in X</math>
If <math>f</math> is continuous, there is an induced cochain map
- <math>f^\sharp:\bar{C}^*(Y;G)\to\bar{C}^*(X;G)</math>
Relative cohomology module
If <math>A</math> is a subspace of <math>X</math> and <math>i:A\hookrightarrow X</math> is an inclusion map, then there is an induced epimorphism <math>i^\sharp:\bar{C}^*(X;G)\to \bar{C}^*(A;G)</math>. The kernel of <math>i^\sharp</math> is a cochain subcomplex of <math>\bar{C}^*(X;G)</math> which is denoted by <math>\bar{C}^*(X,A;G)</math>. If <math>C^*(X,A)</math> denote the subcomplex of <math>C^*(X)</math> of functions <math>\varphi</math> that are locally zero on <math>A</math>, then <math>\bar{C}^*(X,A) = C^*(X,A)/C^*_0(X)</math>.
The relative module is <math>\bar{H}^*(X,A;G)</math> is defined to be the cohomology module of <math>\bar{C}^*(X,A;G)</math>.
<math>\bar{H}^q(X,A;G)</math> is called the Alexander cohomology module of <math>(X,A)</math> of degree <math>q</math> with coefficients <math>G</math> 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 <math>X</math> is a one-point space, <math>G\simeq \bar{H}^*(X;G)</math>
- (Exactness axiom) If <math>(X,A)</math> is a topological pair with inclusion maps <math>i:A\hookrightarrow X</math> and <math>j:X\hookrightarrow (X,A)</math>, there is an exact sequence <math display="block">\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</math>
- (Excision axiom) For topological pair <math>(X,A)</math>, if <math>U</math> is an open subset of <math>X</math> such that <math>\bar{U}\subset\operatorname{int}A</math>, then <math>\bar{C}^*(X,A)\simeq \bar{C}^*(X-U,A-U)</math>.
- (Homotopy axiom) If <math>f_0,f_1:(X,A)\to(Y,B)</math> are homotopic, then <math>f_0^* = f_1^*:H^*(Y,B;G)\to H^*(X,A;G)</math>
Alexander cohomology with compact supports
A subset <math>B\subset X</math> is said to be cobounded if <math>X-B</math> 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 <math>(X,A)</math> by adding the property that <math>\varphi\in C^q(X,A;G)</math> is locally zero on some cobounded subset of <math>X</math>.
Formally, one can define as follows : For given topological pair <math>(X,A)</math>, the submodule <math>C^q_c(X,A;G)</math> of <math>C^q(X,A;G)</math> consists of <math>\varphi\in C^q(X,A;G)</math> such that <math>\varphi</math> is locally zero on some cobounded subset of <math>X</math>.
Similar to the Alexander cohomology module, one can get a cochain complex <math>C^*_c(X,A;G) = \{C^q_c(X,A;G),\delta\}</math> and a cochain complex <math>\bar{C}^*_c(X,A;G) = C^*_c(X,A;G)/C_0^*(X;G)</math>.
The cohomology module induced from the cochain complex <math>\bar{C}^*_c</math> is called the Alexander cohomology of <math>(X,A)</math> with compact supports and denoted by <math>\bar{H}^*_c(X,A;G)</math>. 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 <math>\delta^*:\bar{H}^q_c(A;G)\to \bar{H}^{q+1}_c(X,A;G)</math> only when <math>A\subset X</math> is a closed subset. Similarly, excision axiom can be modified to proper excision axiom i.e. the excision map is a proper map.<ref>Template:Cite book</ref>
Property
One of the most important property of this Alexander cohomology module with compact support is the following theorem:
- If <math>X</math> is a locally compact Hausdorff space and <math>X^+</math> is the one-point compactification of <math>X</math>, then there is an isomorphism <math display="block">\bar{H}^q_c(X;G)\simeq \tilde{\bar{H}}^q(X^+;G).</math>
Example
- <math>\bar{H}^q_c(\R^n;G)\simeq\begin{cases} 0 & q\neq n\\ G & q = n\end{cases}</math>
as <math>(\R^n)^+\cong S^n</math>. Hence if <math>n\neq m</math>, <math>\R^n</math> and <math>\R^m</math> 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<ref>Template:Cite journal</ref> and the first Basic property of tautness, if <math>B\subset A\subset X</math> where <math>X</math> is a paracompact Hausdorff space and <math>A</math> and <math>B</math> are closed subspaces of <math>X</math>, then <math>(A,B)</math> is taut pair in <math>X</math> relative to the Alexander cohomology theory.
Using this tautness property, one can show the following two facts:<ref>Template:Cite book</ref>
- (Strong excision property) Let <math>(X,A)</math> and <math>(Y,B)</math> be pairs with <math>X</math> and <math>Y</math> paracompact Hausdorff and <math>A</math> and <math>B</math> closed. Let <math>f:(X,A)\to(Y,B)</math> be a closed continuous map such that <math>f</math> induces a one-to-one map of <math>X-A</math> onto <math>Y-B</math>. Then for all <math>q</math> and all <math>G</math>, <math display="block">f^*:\bar{H}^q(Y,B;G)\xrightarrow{\sim}\bar{H}^q(X,A;G)</math>
- (Weak continuity property) Let <math>\{(X_\alpha,A_\alpha)\}_\alpha</math> be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let <math display="inline">(X,A) =(\bigcap X_\alpha,\bigcap A_\alpha)</math>. The inclusion maps <math>i_\alpha:(X,A)\to (X_\alpha,A_\alpha)</math> induce an isomorphism
- <math>\{i^*_\alpha\}:\varinjlim\bar{H}^q(X_\alpha,A_\alpha;M)\xrightarrow{\sim}\bar{H}^q(X,A;M)</math>.
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 <math>X</math> is connected if and only if <math>G\simeq \bar{H}^0(X;G)</math>. Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.
If <math>\{U_j\}</math> is an open covering of <math>X</math> by pairwise disjoint sets, then there is a natural isomorphism <math display="inline">\bar{H}^q(X;G)\simeq \prod_j\bar{H}^q(U_j;G)</math>.<ref>Template:Cite book</ref> In particular, if <math>\{C_j\}</math> is the collection of components of a locally connected space <math>X</math>, there is a natural isomorphism <math display="inline">\bar{H}^q(X;G)\simeq \prod_j\bar{H}^q(C_j;G)</math>.
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.