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 $$f,g : M \to N$$ 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 $$\operatorname{Imm}(M, N)$$. Two immersions $$f, g: M \to N$$ are regularly homotopic if they represent points in the same path-component of $$\operatorname{Imm}(M,N)$$.

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 $$\mathbb{R}^n$$ is in one-to-one correspondence with elements of group $$\pi_k\left(V_k\left(\mathbb{R}^n\right)\right)$$. In case $$k = n - 1$$ we have $$V_{n-1}\left(\mathbb{R}^n\right) \cong SO(n)$$. Since $$SO(1)$$ is path connected, $$\pi_2(SO(3)) \cong \pi_2\left(\mathbb{R}P^3\right) \cong \pi_2\left(S^3\right) \cong 0$$ and $$\pi_6(SO(6)) \to \pi_6(SO(7)) \to \pi_6\left(S^6\right) \to \pi_5(SO(6)) \to \pi_5(SO(7))$$ and due to Bott periodicity theorem we have $$\pi_6(SO(6))\cong \pi_6(\operatorname{Spin}(6))\cong \pi_6(SU(4))\cong \pi_6(U(4)) \cong 0$$ and since $$\pi_5(SO(6)) \cong \mathbb{Z},\ \pi_5(SO(7)) \cong 0$$ then we have $$\pi_6(SO(7))\cong 0$$. 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 $$\mathbb{R}^{n+1}$$ 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.