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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 Group s 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 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). BETTI HOMOLOGY AND COHOMOLOGY Since the number of ), 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 Number s of the most familiar spaces such as Simplicial Complex es and Closed Manifold s. |
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