Alexander–Spanier cohomology

It was introduced by {{harvs|txt|authorlink = James Waddell Alexander II|first = James W. |last = Alexander|year = 1935}} for the special case of compact metric spaces, and by {{harvs|txt|authorlink=Edwin Henry Spanier|first=Edwin H.| last=Spanier |year=1948}} for all topological spaces, based on a suggestion of Alexander D. Wallace.

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 $d\backslash colon\; C^\{p-1\}\; \backslash to\; C^\{p\}$ given by

:$df(x\_0,\backslash ldots,x\_p)=\; \backslash sum\_i(-1)^if(x\_0,\backslash ldots,x\_\{i-1\},x\_\{i+1\},\backslash ldots,x\_p).$

The defined cochain complex $C^*(X;G)$ does not rely on the topology of $X$. In fact, if $X$ is a nonempty space, $G\backslash simeq\; H^*(C^*(X;G))$ where $G$ is a graded module whose only nontrivial module is $G$ at degree 0.{{cite book |last1=Spanier |first1=Edwin H. |title=Algebraic topology |date=1966 |page=307 |publisher=Springer |isbn=978-0387944265}}

An element $\backslash varphi\backslash in\; C^p(X)$ is said to be locally zero if there is a covering $\backslash \{U\backslash \}$ of $X$ by open sets such that $\backslash varphi$ vanishes on any $(p+1)$-tuple of $X$ which lies in some element of $\backslash \{U\backslash \}$ (i.e. $\backslash varphi$ vanishes on $\backslash bigcup\_\{U\backslash in\backslash \{U\backslash \}\}U^\{p+1\}$).

The subset of $C^p(X)$ consisting of locally zero functions is a submodule, denote by $C\_0^p(X)$.

$C^*\_0(X)\; =\; \backslash \{C\_0^p(X),d\backslash \}$ is a cochain subcomplex of $C^*(X)$ so we define a quotient cochain complex $\backslash bar\{C\}^*(X)=C^*(X)/C\_0^*(X)$.

The Alexander–Spanier cohomology groups $\backslash bar\{H\}^p(X,G)$ are defined to be the cohomology groups of $\backslash bar\{C\}^*(X)$.

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

:$f^\backslash sharp:C^*(Y;G)\backslash to\; C^*(X;G)$

defined by $(f^\backslash sharp\backslash varphi)(x\_0,...,x\_p)\; =\; (\backslash varphi\; f)(x\_0,...,x\_p),\backslash \; \backslash varphi\backslash in\; C^p(Y);\backslash \; x\_0,...,x\_p\backslash in\; X$

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

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

If $A$ is a subspace of $X$ and $i:A\backslash hookrightarrow\; X$ is an inclusion map, then there is an induced epimorphism $i^\backslash sharp:\backslash bar\{C\}^*(X;G)\backslash to\; \backslash bar\{C\}^*(A;G)$. The kernel of $i^\backslash sharp$ is a cochain subcomplex of $\backslash bar\{C\}^*(X;G)$ which is denoted by $\backslash bar\{C\}^*(X,A;G)$. If $C^*(X,A)$ denote the subcomplex of $C^*(X)$ of functions $\backslash varphi$ that are locally zero on $A$, then $\backslash bar\{C\}^*(X,A)\; =\; C^*(X,A)/C^*\_0(X)$.

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

$\backslash bar\{H\}^q(X,A;G)$ 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

• (Dimension axiom) If $X$ is a one-point space, $G\backslash simeq\; \backslash bar\{H\}^*(X;G)$
• (Exactness axiom) If $(X,A)$ is a topological pair with inclusion maps $i:A\backslash hookrightarrow\; X$ and $j:X\backslash hookrightarrow\; (X,A)$, there is an exact sequence $$\backslash cdots\backslash to\backslash bar\{H\}^q(X,A;G)\; \backslash xrightarrow\{j^*\}\; \backslash bar\{H\}^q(X;G)\backslash xrightarrow\{i^*\}\backslash bar\{H\}^q(A;G)\backslash xrightarrow\{\backslash delta^*\}\backslash bar\{H\}^\{q+1\}(X,A;G)\backslash to\backslash cdots$$
• (Excision axiom) For topological pair $(X,A)$, if $U$ is an open subset of $X$ such that $\backslash bar\{U\}\backslash subset\backslash operatorname\{int\}A$, then $\backslash bar\{C\}^*(X,A)\backslash simeq\; \backslash bar\{C\}^*(X-U,A-U)$.
• (Homotopy axiom) If $f\_0,f\_1:(X,A)\backslash to(Y,B)$ are homotopic, then $f\_0^*\; =\; f\_1^*:H^*(Y,B;G)\backslash to\; H^*(X,A;G)$

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

Similar to the Alexander cohomology module, one can get a cochain complex $C^*\_c(X,A;G)\; =\; \backslash \{C^q\_c(X,A;G),\backslash delta\backslash \}$ and a cochain complex $\backslash bar\{C\}^*\_c(X,A;G)\; =\; C^*\_c(X,A;G)/C\_0^*(X;G)$.

The cohomology module induced from the cochain complex $\backslash bar\{C\}^*\_c$ is called the Alexander cohomology of $(X,A)$ with compact supports and denoted by $\backslash bar\{H\}^*\_c(X,A;G)$. 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 $\backslash delta^*:\backslash bar\{H\}^q\_c(A;G)\backslash to\; \backslash bar\{H\}^\{q+1\}\_c(X,A;G)$ only when $A\backslash subset\; X$ is a closed subset. Similarly, excision axiom can be modified to proper excision axiom i.e. the excision map is a proper map.{{cite book |last1=Spanier |first1=Edwin H. |title=Algebraic topology |date=1966 |isbn=978-0387944265 |pages=320, 322|publisher=Springer }}

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 $$\backslash bar\{H\}^q\_c(X;G)\backslash simeq\; \backslash tilde\{\backslash bar\{H\}\}^q(X^+;G).$$

:

as $(\backslash R^n)^+\backslash cong\; S^n$. Hence if $n\backslash neq\; m$, $\backslash R^n$ and $\backslash R^m$ are not of the same proper homotopy type.

• From the fact that a closed subspace of a paracompact Hausdorff space is a taut subspace relative to the Alexander cohomology theory{{cite journal |last1=Deo |first1=Satya |title=On the tautness property of Alexander-Spanier cohomology |journal=American Mathematical Society |date=197 |volume=52 |pages=441–442}} and the first Basic property of tautness, if $B\backslash subset\; A\backslash subset\; X$ 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:{{cite book |last1=Spanier |first1=Edwin H. |title=Algebraic topology |date=1966 |isbn=978-0387944265 |page=318|publisher=Springer }}

• (Strong excision property) Let $(X,A)$ and $(Y,B)$ be pairs with $X$ and $Y$ paracompact Hausdorff and $A$ and $B$ closed. Let $f:(X,A)\backslash to(Y,B)$ 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 $G$, $$f^*:\backslash bar\{H\}^q(Y,B;G)\backslash xrightarrow\{\backslash sim\}\backslash bar\{H\}^q(X,A;G)$$
• (Weak continuity property) Let $\backslash \{(X\_\backslash alpha,A\_\backslash alpha)\backslash \}\_\backslash alpha$ be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let $(X,A)\; =(\backslash bigcap\; X\_\backslash alpha,\backslash bigcap\; A\_\backslash alpha)$. The inclusion maps $i\_\backslash alpha:(X,A)\backslash to\; (X\_\backslash alpha,A\_\backslash alpha)$ induce an isomorphism
• :$\backslash \{i^*\_\backslash alpha\backslash \}:\backslash varinjlim\backslash bar\{H\}^q(X\_\backslash alpha,A\_\backslash alpha;M)\backslash xrightarrow\{\backslash sim\}\backslash bar\{H\}^q(X,A;M)$.

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

If $\backslash \{U\_j\backslash \}$ is an open covering of $X$ by pairwise disjoint sets, then there is a natural isomorphism $\backslash bar\{H\}^q(X;G)\backslash simeq\; \backslash prod\_j\backslash bar\{H\}^q(U\_j;G)$.{{cite book |last1=Spanier |first1=Edwin H. |title=Algebraic topology |date=1966 |page=310 |publisher=Springer |isbn=978-0387944265}} In particular, if $\backslash \{C\_j\backslash \}$ is the collection of components of a locally connected space $X$, there is a natural isomorphism $\backslash bar\{H\}^q(X;G)\backslash simeq\; \backslash prod\_j\backslash bar\{H\}^q(C\_j;G)$.

It is also possible to define Alexander–Spanier homology{{sfn|Massey|1978a}} and Alexander–Spanier cohomology with compact supports. {{harv|Bredon|1997}}

The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.

{{reflist}}

• {{Citation | last1=Alexander | first1=James W. | author1-link=James Waddell Alexander II | title=On the Chains of a Complex and Their Duals | jstor=86360 | publisher=National Academy of Sciences | year=1935 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=21 | issue=8 | pages=509–511 | doi=10.1073/pnas.21.8.509| pmc=1076641 | bibcode=1935PNAS...21..509A | pmid=16577676| doi-access=free }}
• {{Citation | last1=Bredon | first1=Glen E. | author-link=Glen Bredon | title=Sheaf theory | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-94905-5 | mr=1481706 | year=1997 | volume=170 | doi=10.1007/978-1-4612-0647-7}}
• {{Citation | last1=Massey | first1=William S. | author1-link=William Schumacher Massey | title=How to give an exposition of the Čech-Alexander-Spanier type homology theory | jstor=2321782 | mr=0488017 | year=1978 | journal=The American Mathematical Monthly | issn=0002-9890 | volume=85 | issue=2 | pages=75–83 | doi=10.2307/2321782 |ref={{harvid|Massey|1978a}}}}
• {{Citation | last1=Massey | first1=William S. | author1-link=William Schumacher Massey | title=Homology and cohomology theory. An approach based on Alexander-Spanier cochains. | publisher=Marcel Dekker Inc. | location=New York | series=Monographs and Textbooks in Pure and Applied Mathematics | isbn=978-0-8247-6662-7 | mr=0488016 | year=1978 | volume=46 |ref={{harvid|Massey|1978b}}}}
• {{Citation | last=Spanier | first=Edwin H. | authorlink=Edwin Henry Spanier| title=Cohomology theory for general spaces | jstor=1969289 | mr=0024621 | year=1948 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=49 | issue=2 | pages=407–427 | doi=10.2307/1969289 }}
• {{Citation |last1=Spanier |first1=Edwin H. | authorlink=Edwin Henry Spanier |title=Algebraic topology |date=1966 |publisher=Springer |isbn=978-0387944265}}

{{DEFAULTSORT:Alexander-Spanier cohomology}}

Category:Cohomology theories

Category:Duality theories