Alexander–Spanier cohomology

It was introduced by for the special case of compact metric spaces, and by 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)^if(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 \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^q_c(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^q_c(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}^q_c(A;G)\to \bar{H}^{q+1}_c(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}^q_c(X;G)\simeq \tilde{\bar{H}}^q(X^+;G). Example :\bar{H}^q_c(\R^n;G)\simeq\begin{cases} 0 & q\neq n\\ G & q = n\end{cases} as (\R^n)^+\cong S^n. Hence if n\neq 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^*:\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) =(\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 \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 \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. ==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==