Homotopic

In topology, two continuous maps f and g from a space X to a space Y are said to be homotopic if there exists a continuous map H: X × [0,1] -> Y, called a homotopy, such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. The map H is a homotopy from f to g, and f and g are said to lie in the same homotopy class.

Homotopy defines an equivalence relation on the set of continuous maps X -> Y: being in the same homotopy class means the two maps can be continuously deformed into one another by a parameter t in [0,1]. The collection of all maps from X to Y modulo homotopy is central to the study of a space’s “homotopy type.”

Spaces X and Y are called homotopy equivalent if there exist maps f: X -> Y and g: Y -> X such that the compositions gf and fg are each homotopic to the respective identity maps on Y and X. Homotopy equivalence identifies spaces that can be deformed into one another through continuous mappings, preserving many topological properties while ignoring finer geometric details.

Homotopy is a homotopy-invariant concept; many invariants, such as fundamental groups and higher homotopy groups, are preserved under homotopy. For example, maps from the n-sphere to itself are classified up to homotopy by their degree, an integer-valued invariant.

The notion of homotopy generalizes to relative homotopy, where deformations are fixed on a subspace A ⊆ X, and to families of maps, yielding a powerful framework for classifying spaces up to deformation rather than rigid equivalence.