Exact sequences

At each position, the incoming image equals the outgoing kernel.

Different ways to understand it

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Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Higher homotopy groups
Based maps from spheres modulo homotopy measure higher-dimensional homotopy structure.
Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Covering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.
Euler characteristic
Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.