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

Examples

This curve has total curvature 6π, and turning number 3.

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.

Smale's classification of immersions of spheres shows that sphere eversions exist, which can be realized via this Morin surface.

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. 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.

References

Hassler Whitney, On regular closed curves in the plane. Compositio Mathematica, 4 (1937), p. 276–284
Stephen Smale, A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281–290.
Stephen Smale, The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69 1959 327–344.