Regular homotopy

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions formula_1 are homotopic if they represent points in the same path-components of the mapping space formula_2, given the compact-open topology. The space of immersions is the subspace of formula_2 consisting of immersions, denoted by formula_4. Two immersions formula_5 are regularly homotopic if they represent points in the same path-component of formula_4.
Examples.
The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if their Gauss maps have the same degree/winding number.
Stephen Smale classified the regular homotopy classes of a "k"-sphere immersed in formula_7 – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here "k" partial derivatives not vanishing. More precisely, the set formula_8 of regular homotopy classes of embeddings of sphere formula_9 in formula_10 is in one-to-one correspondence with elements of group formula_11. In case formula_12 we have formula_13. Since formula_14 is path connected, formula_15 and formula_16 and due to Bott periodicity theorem we have formula_17 and since formula_18 then we have formula_19. Therefore all immersions of spheres formula_20 and formula_21 in euclidean spaces of one more dimension are regular homotopic. In particular, spheres formula_22 embedded in formula_23 admit eversion if formula_24. A corollary of his work is that there is only one regular homotopy class of a "2"-sphere immersed in formula_25. In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".
Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the homotopy principle (or "h"-principle) approach.