Alexander–Spanier cohomology

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

History

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.

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 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 homomorphism

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 module

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 axioms

  • (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 supports

A subset is said to be cobounded if 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 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]

Property

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

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

Example

as . Hence if , and 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[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 isomorphism
    .

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 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 .

Variants

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

References

  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.

Bibliography

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