Regular homotopy

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 ${\displaystyle \mathbb {R} ^{n}}$  – they are classified by homotopy groupsofStiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing. More precisely, the set ${\displaystyle I(n,k)}$  of regular homotopy classes of embeddings of sphere ${\displaystyle S^{k}}$ in${\displaystyle \mathbb {R} ^{n}}$  is in one-to-one correspondence with elements of group ${\displaystyle \pi _{k}\left(V_{k}\left(\mathbb {R} ^{n}\right)\right)}$ . In case ${\displaystyle k=n-1}$  we have ${\displaystyle V_{n-1}\left(\mathbb {R} ^{n}\right)\cong SO($n)}  . Since ${\displaystyle SO($1)}   is path connected, ${\displaystyle \pi _{2}(SO($3))\cong \pi _{2}\left(\mathbb {R} P^{3}\right)\cong \pi _{2}\left(S^{3}\right)\cong 0}   and ${\displaystyle \pi _{6}(SO($6))\to \pi _{6}(SO(7))\to \pi _{6}\left(S^{6}\right)\to \pi _{5}(SO(6))\to \pi _{5}(SO(7))}   and due to Bott periodicity theorem we have ${\displaystyle \pi _{6}(SO($6))\cong \pi _{6}(\operatorname {Spin} (6))\cong \pi _{6}(SU(4))\cong \pi _{6}(U(4))\cong 0}   and since ${\displaystyle \pi _{5}(SO($6))\cong \mathbb {Z} ,\ \pi _{5}(SO(7))\cong 0}   then we have ${\displaystyle \pi _{6}(SO($7))\cong 0}  . Therefore all immersions of spheres ${\displaystyle S^{0},\ S^{2}}$  and ${\displaystyle S^{6}}$  in euclidean spaces of one more dimension are regular homotopic. In particular, spheres ${\displaystyle S^{n}}$  embedded in ${\displaystyle \mathbb {R} ^{n+1}}$  admit eversion if ${\displaystyle n=0,2,6}$ . 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 (orh-principle) approach.

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]

• 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.