What is homology in algebraic topology?

Homology is a fundamental concept in algebraic topology used to study topological spaces through sequences of abelian groups called homology groups. Originally introduced to capture topological ideas like holes and cycles, it acts as an invariant associated with chain complexes. Under suitable conditions, different homology theories yield a consistent single homology for a space.

What we know

• Homology was originally introduced in algebraic topology and involves sequences of abelian groups called homology groups. [3]
• Homology is closely related to topological features frequently discussed in popular mathematics, such as the holes of a surface or the cycles of a graph. [3]
• A homology sphere is an n-manifold that shares the homology groups of an n-sphere for an integer n greater than or equal to 1. [4]
• Homological algebra studies homology in a general algebraic setting, focusing on chain complexes, homological functors, and homological invariants. [5]

Where to go next

• the bigger pictureHow does homological algebra study homology generally?
• a case in pointWhat is a homology sphere in topology?
• how it workedHow are chain complexes used to define homology groups?
• compared withHow does algebraic topology differ from geometric topology?
• what followedWhat are algebraic invariants in topological spaces?
• an unexpected connectionHow does topology help solve abstract algebra problems?Although algebraic topology primarily uses algebra to study topology, the field can also be turned around to provide convenient proofs for purely algebraic problems like showing any subgroup of a free group is free.

Sources

• [1]algebraic topology (Wikidata Q212803) · CC0 1.0
• [2]Algebraic topology · CC BY-SA 4.0
• [3]Homology (mathematics) · CC BY-SA 4.0
• [4]Homology sphere · CC BY-SA 4.0
• [5]Homological algebra · CC BY-SA 4.0

Your path

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