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 $f,g:M\to N$ 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 $\mathrm{Imm}(M,N)$. Two immersions $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\mathrm{Imm}(M,N)$.

Examples

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

File:Winding Number Around Point.svg

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.

File:MorinSurfaceFromTheTop.PNG

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 ${\mathbb{ℝ}}^n$ – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing. More precisely, the set $I(n,k)$ of regular homotopy classes of embeddings of sphere $S^k$ in ${\mathbb{ℝ}}^n$ is in one-to-one correspondence with elements of group ${\pi }_k\left(V_k\left({\mathbb{ℝ}}^n\right)\right)$. In case $k=n-1$ we have $V_{n-1}\left({\mathbb{ℝ}}^n\right)\cong SO(n)$. Since $SO(1)$ is path connected, ${\pi }_2(SO(3))\cong {\pi }_2\left(\mathbb{ℝ}{P}^3\right)\cong {\pi }_2\left(S^3\right)\cong 0$ and ${\pi }_6(SO(6))\to {\pi }_6(SO(7))\to {\pi }_6\left(S^6\right)\to {\pi }_5(SO(6))\to {\pi }_5(SO(7))$ and due to Bott periodicity theorem we have ${\pi }_6(SO(6))\cong {\pi }_6(\mathrm{Spin}(6))\cong {\pi }_6(SU(4))\cong {\pi }_6(U(4))\cong 0$ and since $$\begin{gathered} {\pi }_5(SO(6))\cong \mathbb{\mathbb{Z}}, \\ {\pi }_5(SO(7))\cong 0 \end{gathered}$$ then we have ${\pi }_6(SO(7))\cong 0$. Therefore all immersions of spheres $$\begin{gathered} S^0, \\ {S}^2 \end{gathered}$$ and $S^6$ in euclidean spaces of one more dimension are regular homotopic. In particular, spheres $S^n$ embedded in ${\mathbb{ℝ}}^{n+1}$ 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 ${\mathbb{ℝ}}^3$. 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.

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.

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.