Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory for more background.

Properties

If (dn : An -> An-1) is a chain complex such that all but finitely many An are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the Euler characteristic

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:χ = ∑ (-1)n rank(An)

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(using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:

Related Topics:
Rank - Hamel dimension

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:χ = ∑ (-1)n rank(Hn)

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and, especially in algebraic topology, this provides two ways to compute the important invariant χ for the object X which gave rise to the chain complex.

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Every short exact sequence

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:0 -> A -> B -> C -> 0

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of chain complexes gives rise to a long exact sequence of homology groups

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: ... -> Hn(A) -> Hn(B) -> Hn(C) -> Hn-1(A) -> Hn-1(B) -> Hn-1(C) -> Hn-2(A) -> ...

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All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps Hn(C) -> Hn-1(A). These latter are called connecting homomorphisms and are provided by the snake lemma.

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