Alexander-Spanier cohomology
Encyclopedia
In mathematics
, particularly in algebraic topology
, Alexander–Spanier cohomology is a cohomology
theory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D. Wallace.
It is also possible to define Alexander–Spanier homology and Alexander–Spanier cohomology with compact supports .
The Alexander–Spanier cohomology groups coincide with Cech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.
there is a complex C whose pth term Cp is the set of (possibly discontinuous) functions from Xp+1 to G with differential d given by
It has a subcomplex C0 of functions that vanish in a neighborhood of the diagonal. The Alexander–Spanier cohomology groups Hp(X,G) are defined to be the cohomology groups of the complex C/C0.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, Alexander–Spanier cohomology is a cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
theory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D. Wallace.
It is also possible to define Alexander–Spanier homology and Alexander–Spanier cohomology with compact supports .
The Alexander–Spanier cohomology groups coincide with Cech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.
Definition
If X is a topological space and G is an abelian group, thenthere is a complex C whose pth term Cp is the set of (possibly discontinuous) functions from Xp+1 to G with differential d given by
It has a subcomplex C0 of functions that vanish in a neighborhood of the diagonal. The Alexander–Spanier cohomology groups Hp(X,G) are defined to be the cohomology groups of the complex C/C0.