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

Examples

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.

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 $\pi _{k}(V_{k}(\mathbb {R} ^{n}))$ . In case $k=n-1$ we have $V_{n-1}(\mathbb {R} ^{n})\cong SO(n)$ . Since $SO(1)$ is path connected, $\pi _{2}(SO(3))\cong \pi _{2}(\mathbb {R} P^{3})\cong \pi _{2}(S^{3})\cong 0$ and $\pi _{6}(SO(6))\to \pi _{6}(SO(7))\to \pi _{6}(S^{6})\to \pi _{5}(SO(6)\to \pi _{5}(SO(7))$ 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 $\pi _{5}(SO(6))\cong \mathbb {Z} ,\ \pi _{5}(SO(7))\cong 0$ 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 $\mathbb {R} ^{n+1}$ 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.

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.

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