Singular homology

See also: Homology

In mathematics, the singular homology one of constructions which make it possible to calculate the Homologie of a topological Espace, is named X thereafter. It is calculated starting from a Complexe chains whose terms of index N is the free abelian Groupe generated by the continuous applications of the standard Simplexe $\ Delta_n$ of dimension N towards space X . The application of edge of the complex of chain is that which with an application towards X of source N - standard simplex (directed) associates the alternate sum of its restrictions on the various edges of $\ Delta_n$ , all identified with the simplex of dimension n-1 .

In general the built complex is very large and incalculable in practice. For example, the first group, of index zero, is the group of the formal sums, with relative whole coefficients, of the points of studied space: it is a free abelian group of row the cardinal of X . N the other hand, the singular homology offers a great flexibility as for the choice of spaces considered.

However the groups of homology which result some have the properties of finitude discounted for spaces X reasonable. One indeed has theorems of comparisons with other constructions of these groups of homology.

The complex

Results

Properties

Generalizations

Let us mention finally that methods inspired of the singular homology are applied in algebraic Géométrie, within the framework of the homotopic theories of the diagrams. The purpose of they are to define a motivic Cohomologie, and have spectacular repercussions in Arithmétique.

Internal bonds

Homology and cohomology
Zéroième groups cellular homology
Homologie
Homologie simpliciale
Homologie of the groups

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