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Regular homotopy

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

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Examples

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Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy.

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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.

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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 – 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 of regular homotopy classes of embeddings of sphere in is in one-to-one correspondence with elements of group . In case we have . Since is path connected, and and due to Bott periodicity theorem we have and since then we have . Therefore all immersions of spheres and in euclidean spaces of one more dimension are regular homotopic. In particular, spheres embedded in admit eversion if , 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.

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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.[1] Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.[2]

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References

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