Regular homotopy

 ( Differential Topology ) 

In the mathematics 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 , 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 $f,g\; :\; M\; \backslash to\; N$ 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 subset of $C(M,\; N)$ consisting of immersions, denoted by $\backslash operatorname\{Imm\}(M,\; N)$, given the $C^1$ topology. Two immersions $f,\; g:\; M\; \backslash to\; N$ are regularly homotopic if they represent points in the same path-component of $\backslash operatorname\{Imm\}(M,N)$.

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Examples Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. 6 π, 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 have the same degree/winding number.

Stephen Smale classified the regular homotopy classes of a k-sphere immersed in $\backslash mathbb\; R^n$ – they are classified by homotopy groups of , 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 $\backslash mathbb\{R\}^n$ is in one-to-one correspondence with elements of group $\backslash pi\_k\backslash left(V\_k\backslash left(\backslash mathbb\{R\}^n\backslash right)\backslash right)$. In case $k\; =\; n\; -\; 1$ we have $V\_\{n-1\}\backslash left(\backslash mathbb\{R\}^n\backslash right)\; \backslash cong\; SO(n)$. Since $SO(1)$ is path connected, $\backslash pi\_2(SO(3))\; \backslash cong\; \backslash pi\_2\backslash left(\backslash mathbb\{R\}P^3\backslash right)\; \backslash cong\; \backslash pi\_2\backslash left(S^3\backslash right)\; \backslash cong\; 0$ and $\backslash pi\_6(SO(6))\; \backslash to\; \backslash pi\_6(SO(7))\; \backslash to\; \backslash pi\_6\backslash left(S^6\backslash right)\; \backslash to\; \backslash pi\_5(SO(6))\; \backslash to\; \backslash pi\_5(SO(7))$ and due to Bott periodicity theorem we have $\backslash pi\_6(SO(6))\backslash cong\; \backslash pi\_6(\backslash operatorname\{Spin\}(6))\backslash cong\; \backslash pi\_6(SU(4))\backslash cong\; \backslash pi\_6(U(4))\; \backslash cong\; 0$ and since $\backslash pi\_5(SO(6))\; \backslash cong\; \backslash mathbb\{Z\},\backslash \; \backslash pi\_5(SO(7))\; \backslash cong\; 0$ then we have $\backslash pi\_6(SO(7))\backslash cong\; 0$. Therefore all immersions of spheres $S^0,\backslash \; S^2$ and $S^6$ in euclidean spaces of one more dimension are regular homotopic. In particular, spheres $S^n$ embedded in $\backslash mathbb\{R\}^\{n+1\}$ admit sphere eversion if $n\; =\; 0,\; 2,\; 6$, 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 , 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.

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See also

• Arnold invariants

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