Homotopy equivalence

Maps with inverse compositions homotopic to identities describe spaces with the same homotopy type.

Different ways to understand it

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Covering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.
Homotopy
A homotopy continuously deforms one map into another.
Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Euler characteristic
Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.