Abstract
Here we first meet the basic notions of homology in simple geometric cases where the homology group arises from a boundary operator. In general, an abelian group with a boundary operator is called a “differential group” or, when provided with dimensions, a “chain complex”. This chapter considers the algebraic process of constructing homology and cohomology groups from chain complexes. Basic is the fact (§ 4) that a short exact sequence of complexes gives a long exact sequence of homology groups. As illustrative background, the last sections provide a brief description of the singular homology groups of a topological space.
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© 1995 Springer-Verlag Berlin Heidelberg
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Mac Lane, S. (1995). Homology of Complexes. In: Homology. Classics in Mathematics, vol 114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-62029-4_3
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DOI: https://doi.org/10.1007/978-3-642-62029-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58662-3
Online ISBN: 978-3-642-62029-4
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