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 are homotopic if they represent points in the same path-components of the mapping space C(M, N), given the compact-open topology. The space of immersions is the subspace of C(M, N) consisting of immersions, denoted by. Two immersions are regularly homotopic if they represent points in the same path-component of.

Examples

Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. 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 \mathbb R^n – 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 I(n,k) of regular homotopy classes of embeddings of sphere S^k in is in one-to-one correspondence with elements of group. In case k = n - 1 we have. Since SO(1) is path connected, and and due to Bott periodicity theorem we have and since then we have. Therefore all immersions of spheres S^0,\ S^2 and S^6 in euclidean spaces of one more dimension are regular homotopic. In particular, spheres S^n embedded in admit eversion if n = 0, 2, 6. A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in \mathbb R^3. 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.

Non-degenerate homotopy

For locally convex, closed space curves, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish). There are 2 distinct non-degenerate homotopy classes. Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.