Homology theory

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.

Related Topics:
Mathematics - Axiom - Topological space - Homology

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At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher. The simplest case is in graph theory, with C and D vertices and homology with a meaning coming from the oriented edge E from P to Q having boundary Q — P. A collection of edges from D to C, each one joining up to the one before, is a homology. In general, a k-chain is thought of as a formal combination

Related Topics:
Equivalence relation - Graph theory

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:Σ aidi

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where the ai are integers and the di are k-dimensional simplices on X. The boundary concept here is that taken over from the boundary of a simplex; it allows a high-dimensional concept which for k = 1 is the kind of telescopic cancellation seen in the graph theory case. This explanation is in the style of 1900, and proved somewhat naive, technically speaking.

Related Topics:
Simplices - Telescopic cancellation

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