Homotopy

Homotopy is a relation between continuous maps that formalizes the idea of deforming one function into another through a family of intermediate maps. Let X and Y be topological spaces and f, g: X → Y be continuous. A homotopy from f to g is a continuous map H: X × [0,1] → Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. If a subspace A ⊂ X or a basepoint is fixed during the deformation, the homotopy is called relative to A or based, respectively.

Homotopy classes group maps into equivalence classes: two maps are homotopic if there is a homotopy joining them. The set of based homotopy classes is denoted [X, Y] when X is based and maps preserve the basepoint. A stronger notion is homotopy equivalence: spaces X and Y have the same homotopy type if there exist continuous maps f: X → Y and g: Y → X such that g ∘ f is homotopic to the identity on X and f ∘ g is homotopic to the identity on Y. Homotopy equivalence identifies spaces that cannot be distinguished by homotopy-invariant properties.

A central construction is the fundamental group π1(X, x0), formed from based loops at x0 up to based homotopy, with the loop concatenation operation. Higher homotopy groups πn(X, x0) for n ≥ 2, defined via based maps from the n-sphere S^n, are abelian for n ≥ 2. These groups are fundamental tools in algebraic topology, encoding information about a space’s shape that is preserved under homotopy.

Other related notions include deformation retracts (subspaces that retain the ambient space’s homotopy type) and relative homotopy theory (maps that agree on a subspace). The study of homotopy leads to concepts such as homotopy types, CW complexes, and the homotopy category, and underpins many classification and computation techniques in topology.

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