Isotopy

Isotopy is a concept in topology and related fields that formalizes when one geometric object can be continuously deformed into another within a fixed ambient space, while preserving designated structural properties throughout the deformation.

In topology, an isotopy between embeddings f0,f1: X→Y is a continuous map F:X×I→Y with F(x,0)=f0(x), F(x,1)=f1(x), and such that for each t∈I, the map x↦F(x,t) is an embedding. If the ambient space Y is kept fixed by a family of homeomorphisms H_t:Y→Y with H_0=id_Y and f1=H_1∘f0, the embeddings are ambient isotopic.

In differential topology, isotopy is often defined for diffeomorphisms. A smooth isotopy between f0,f1:M→N is a smooth F:M×I→N with F(x,0)=f0(x), F(x,1)=f1(x) and each t gives a diffeomorphism. If M=N and all F_t are diffeomorphisms, the isotopy is a path in the diffeomorphism group Diff(M).

In knot theory, two knots are considered equivalent if they are ambient isotopic as embeddings of S^1 into R^3, meaning there is an ambient isotopy of R^3 taking one knot to the other.

In algebra, isotopy describes a relation between algebraic structures such as quasigroups or loops. An isotopy consists of bijections α,β,γ with α(x)⋆β(y)=γ(x⋆y); isotopic structures need not be isomorphic, but share a related multiplication table up to these maps.

Isotopy is stronger than homotopy because intermediate objects must retain embedding or homeomorphism properties. The isotopy extension theorem provides conditions under which a deformation of a submanifold extends to an ambient isotopy of the ambient space.