Homology groups

Cycles modulo boundaries measure holes through a chain complex.

Different ways to understand it

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Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Euler characteristic
Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.
Persistent homology
Homological features are tracked across a nested family of spaces or data scales.
Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Covering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.