Alexander–Spanier cohomology

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

HistoryEdit

It was introduced by James W. Alexander (1935) for the special case of compact metric spaces, and by Edwin H. Spanier (1948) for all topological spaces, based on a suggestion of Alexander D. Wallace.

DefinitionEdit

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   is the set of all functions from   to G with differential   given by

 

The defined cochain complex   does not rely on the topology of  . In fact, if   is a nonempty space,   where   is a graded module whose only nontrivial module is   at degree 0.[1]

An element   is said to be locally zero if there is a covering   of   by open sets such that   vanishes on any  -tuple of   which lies in some element of   (i.e.   vanishes on  ). The subset of   consisting of locally zero functions is a submodule, denote by  .   is a cochain subcomplex of   so we define a quotient cochain complex  . The Alexander–Spanier cohomology groups   are defined to be the cohomology groups of  .

Induced homomorphismEdit

Given a function   which is not necessarily continuous, there is an induced cochain map

 

defined by  

If   is continuous, there is an induced cochain map

 

relative cohomology moduleEdit

If   is a subspace of   and   is an inclusion map, then there is an induced epimorphism  . The kernel of   is a cochain subcomplex of   which is denoted by  . If   denote the subcomplex of   of functions   that are locally zero on  , then  .

The relative module is   is defined to be the cohomology module of  

  is called the Alexander cohomology module of   of degree   with coefficients   and this module satisfies all cohomology axioms. The resulting cohomology theory is called the Alexander (or Alexander-Spanier) cohomology theory

Cohomology theory axiomsEdit

  • (Dimension axiom) If   is a one-point space,  
  • (Exactness axiom) If   is a topological pair with inclusion maps   and  , there is an exact sequence
     
  • (Excision axiom) For topological pair  , if   is an open subset of   such that  , then  .
  • (Homotopy axiom) If   are homotopic, then  

Alexander cohomology with compact supportsEdit

A subset   is said to be cobounded if   is bounded.

Similar to the definition of Alexander cohomology module, one can define Alexander cohomology module with compact supports of a pair   by adding the property that   is locally zero on some cobounded subset of  .

Formally, one can define as follows : For given topological pair  , the submodule   of   consists of   such that   is locally zero on some cobounded subset of  .

Similar to the Alexander cohomology module, one can get a cochain complex   and a cochain complex  .

The cohomology module induced from the cochain complex   is called the Alexander cohomology of   with compact supports and denoted by  . 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   only when   is a closed subset. Similarly, excision axiom can be modified to proper excision axiom i.e. the excision map is a proper map.[2]

PropertyEdit

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

  • If   is locally compact Hausdorff space and   is the one-point compactification of  , there is an isomorphism
     

ExampleEdit

 

as  . Hence if  ,   and   are not of the same proper homotopy type.

Relation with tautnessEdit

  • From the fact that a closed subspace of a paracompact Hausdorff space is a taut subspace relative to the Alexander cohomology theory [3] and the first Basic property of tautness, if   where   is a paracompact Hausdorff space and   and   are closed subspaces of  , then   is taut pair in   relative to the Alexander cohomology theory.

Using this tautness property, one can show the following two facts[4]

  • (Strong excision property) Let   and   be pairs with   and   paracompact Hausdorff and   and   closed. Let   be a closed continuous map such that   induces a one-to-one map of   onto  . Then for all   and all  ,
     
  • (Weak continuity property) Let   be a family of compact Hausdorff pairs in some space, directed downward by inclusion, and let  . The inclusion maps   induce an isomoprhim
 

Difference from singular cohomology theoryEdit

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   is connected if and only if  . Hence for any connected space which is not path connected, singular cohomology and Alexander cohomology differ in degree 0.

If   is an open covering of   by pairwise disjoint sets, then there is a natural isomorphism  .[5] In particular, if   is the collection of components of a locally connected space  , there is a natural isomorphism  

VariantsEdit

It is also possible to define Alexander–Spanier homology,.[6]

Connection to other cohomologiesEdit

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

ReferencesEdit

  1. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 307. ISBN 978-0387944265.
  2. ^ Spanier, Edwin H. (1966). Algebraic topology. pp. 320, 322. ISBN 978-0387944265.
  3. ^ Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American Mathematical Society. 52: 441–442.
  4. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 318. ISBN 978-0387944265.
  5. ^ Spanier, Edwin H. (1966). Algebraic topology. p. 310. ISBN 978-0387944265.
  6. ^ Massey 1978a.

BibliographyEdit