What is chain complex homology in mathematics?

In mathematics, a chain complex consists of a sequence of modules or abelian groups connected by homomorphisms where the image of each map is contained in the kernel of the next. Associated to this structure is its homology, which loosely measures the failure of the chain complex to be exact. Homology groups serve as fundamental algebraic invariants used widely across algebraic topology and abstract algebra.

What we know

• A chain complex is an algebraic structure consisting of a sequence of abelian groups or modules and homomorphisms between consecutive groups where the image of each homomorphism is contained in the kernel of the next. [1, 2]
• The homology of a chain complex is a measure of the failure of the chain complex to be exact. [2]
• Homology groups are regarded as fundamental invariants of a chain complex and can be associated to different mathematical objects through specific procedures. [3]
• Homological algebra is the branch of mathematics that studies homology in a general algebraic setting, tracing its origins to combinatorial topology and abstract algebra. [5]

Where to go next

• compared withWhat is a cochain complex?
• a case in pointHow does simplicial homology work?
• the bigger pictureWhat is homological algebra?
• how we knowHow does singular homology study topological spaces?
• how it workedWhat is a spectral sequence?
• an unexpected connectionHow does homological algebra connect to mathematical physics?Although homological algebra originated from abstract algebra and topology, its influence has expanded to include mathematical physics.

Sources

• [1]chain complex (Wikidata Q1143328) · CC0 1.0
• [2]Chain complex · CC BY-SA 4.0
• [3]Homology (mathematics) · CC BY-SA 4.0
• [4]Simplicial homology · CC BY-SA 4.0
• [5]Homological algebra · CC BY-SA 4.0

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