Singular homology

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In algebraic topology, a branch of mathematics, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. (Alternatively, R-modules and R-linear homomorphisms can be used instead of abelian groups and group homomorphisms.)

The homology of a space X is usually understood to mean the singular homology of that space.

Singular homology is constructed by applying the general homology construction to the singular chain complex, the chain complex of formal sums of singular simplices.

1 Singular simplices
2 Singular chain complex
3 Coefficients in R
4 Cohomology
5 Betti homology and cohomology

[edit] Singular simplices

A singular n-simplex is a continuous mapping σ from the standard n-simplex to a topological space X. This mapping need not be injective, and there can be non-equivalent singular simplices with the same image in X.

The boundary of σ, dσ, is defined to be the formal sum of the singular (n−1)-simplices represented by the restriction of σ to the faces of the standard n-simplex, with an alternating sign to take orientation into account. In particular, the boundary of a 1-simplex $σ$ is the formal difference $σ 1 - σ 0$ .

[edit] Singular chain complex

If we consider the free abelian groups generated by all singular n-simplices and extend the boundary operator $d$ to formal sums of singular $n$ -simplices, we obtain a chain complex of abelian groups.

The $n$ -th homology group of $X$ is then defined as the factor group

$H_{n}(X) = \frac{\ker (d_{n})}{\mathrm{im} (d_{n+1})}$ .

[edit] Coefficients in R

If R is any ring (assumed unital on Wikipedia), we can replace free abelian groups by free R-modules. The definition of d does not change, but H_n(X, R) now is an R-module (not necessarily free).

[edit] Cohomology

By dualizing the homology chain complex (i.e. applying the functor Hom(-, R), R being any ring) we obtain a cochain complex with coboundary map $δ$ . The cohomology groups of X are defined as the cohomology groups of this complex. They form a graded R-module, which can be given the structure of a graded R-algebra using the cup product.

[edit] Betti homology and cohomology

Since the number of homology theories has become large (see Category:Homology theory), the terms Betti homology and Betti cohomology are sometimes applied (particularly by authors writing on algebraic geometry), to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.

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Category: Homology theory