Isotopy
In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism.
Requiring that two homeomorphisms be isotopic is a stronger requirement than that they be homotopic. For example, the map on the interval in R defined by f(x) = −x is not isotopic to the identity g(x) = x. Any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed the orientation of the interval and g has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from f to the identity is H: × → given by H(x,y) = 2yx-x.
Two homeomorphisms of the unit ball which agree on the boundary can be shown to be isotopic using Alexander's trick. For this reason, the map of the unit disc in R2 defined by f(x,y) = (−x, −y) is isotopic to a 180-degree rotation around the origin, and so the identity map and f are isotopic because they can be connected by rotations.
In geometric topology—for example in knot theory—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots, K1 and K2, in three-dimensional space. A knot is an embedding of a one-dimensional space, the "loop of string", into this space, and an embedding is simply a homeomorphism. The intuitive idea of deforming one to the other should correspond to a path of embeddings: a continuous function starting at t=0 with the K1 embedding, ending at t=1 with the K2 embedding, with all intermediate values being embeddings; this corresponds to the definition of isotopy. This works well as long as we require all the maps involved to be differentiable, but fails if they are simply allowed to be continuous, as there is then no obstruction to pulling the knot taut, which changes the knot geometry. An ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K1 and K2 are considered equivalent when there is an ambient isotopy which moves K1 to K2. This is the appropriate definition in the topological category.
Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example a path between two diffeomorphisms 'through diffeomorphism space' is a smooth isotopy, or a path through symplectomorphisms is a symplectic isotopy.
Read more about this topic: Homotopy