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 ${\displaystyle f,g:M\to N}$ are homotopic if they represent points in the same path-components of the mapping space ${\displaystyle C(M,N)}$, given the compact-open topology. The space of immersions is the subspace of ${\displaystyle C(M,N)}$ consisting of immersions, denote it by ${\displaystyle Imm(M,N)}$. Two immersions ${\displaystyle f,g:M\to N}$ are regularly homotopic if they represent points in the same path-component of ${\displaystyle Imm(M,N)}$.

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 ${\displaystyle \mathbb {R} ^{n}}$ – 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 ${\displaystyle \mathbb {R} ^{3}}$. 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

• Whitney, Hassler (1937). "On regular closed curves in the plane". Compositio Mathematica. 4: 276–284.
• Smale, Stephen (February 1959). "A classification of immersions of the two-sphere" (PDF). Transactions of the American Mathematical Society. 90 (2): 281–290. doi:10.2307/1993205. JSTOR 1993205.
• Smale, Stephen (March 1959). "The classification of immersions of spheres in Euclidean spaces" (PDF). Annals of Mathematics. 69 (2): 327–344. doi:10.2307/1970186. JSTOR 1970186.