Difference between revisions of "Differential group"
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An [[Abelian group]] $C$ with a given endomorphism $d : C \rightarrow C$ such that $d^2 = 0$. This endomorphism is called a ''differential''. The elements of a differential group are known as chains; the elements of the kernel $\ker d$ are known as ''cycles''; and the elements of the image $\mathrm{im}\, d$ are called ''boundaries''. The ''homology'' of $C$ is the quotient $\ker d / \mathrm{im}\,d$. | An [[Abelian group]] $C$ with a given endomorphism $d : C \rightarrow C$ such that $d^2 = 0$. This endomorphism is called a ''differential''. The elements of a differential group are known as chains; the elements of the kernel $\ker d$ are known as ''cycles''; and the elements of the image $\mathrm{im}\, d$ are called ''boundaries''. The ''homology'' of $C$ is the quotient $\ker d / \mathrm{im}\,d$. | ||
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| − | + | * {{Ref|a1}} E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. 156 | |
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Latest revision as of 14:52, 8 April 2023
An Abelian group $C$ with a given endomorphism $d : C \rightarrow C$ such that $d^2 = 0$. This endomorphism is called a differential. The elements of a differential group are known as chains; the elements of the kernel $\ker d$ are known as cycles; and the elements of the image $\mathrm{im}\, d$ are called boundaries. The homology of $C$ is the quotient $\ker d / \mathrm{im}\,d$.
References
- [a1] E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. 156
How to Cite This Entry:
Differential group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_group&oldid=53679
Differential group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_group&oldid=53679
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article