Regular homotopy explained

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

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 R ^{n} is in one-to-one correspondence with elements of group \pi _{k\left(V} ^{n\right)\right)} ^{k\left( R} . In case k = n - 1 we have V _{n-1} \left( R ^{n\right)} \cong SO(n) . Since SO( 1 ) is path connected, \pi _{2(SO(3))} \cong \pi ^{3 \right)} _{2 \left( R P} \cong \pi ^{3 \right)} _{2 \left(S} \cong 0 and \pi _{6(SO(6))} \to \pi _{6(SO(7))} \to \pi ^{6 \right)} _{6 \left(S} \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 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 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 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.^[1] Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.^[2]

References

• Hassler . Whitney . Hassler Whitney . On regular closed curves in the plane . . 4 . 1937 . 276–284 .
• Stephen . Smale . Stephen Smale . A classification of immersions of the two-sphere . . 90 . 2 . February 1959 . 281–290 . 1993205 . 10.2307/1993205 . free .
• Stephen . Smale . Stephen Smale . The classification of immersions of spheres in Euclidean spaces . . 2 . 69 . March 1959 . 327–344 . 1970186 . 10.2307/1970186 .

Notes and References

1. Feldman. E. A.. 1968. Deformations of closed space curves. Journal of Differential Geometry. en. 2. 1. 67–75. 10.4310/jdg/1214501138 . free.
2. Little. John A.. 1971. Third order nondegenerate homotopies of space curves. Journal of Differential Geometry. en. 5. 3. 503–515. 10.4310/jdg/1214430012 . free.