Singular homology
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In algebraic topology, 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.
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.
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.
Thus, in particular, the boundary of a 1-simplex σ is the formal difference
- σ(1) − σ(0).
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
- Hn(X) = ker(dn) / im(dn+1).
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 Hn(X, R) now is an R-module (not necessarily free).