In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersionss 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, given the compact-open topology. The space of immersions are regularly homotopic.

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 – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k and due to Bott periodicity theorem admit eversion, i.e. one can turn these spheres "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.

See also

• Arnold invariants

References