Regular homotopy
From Wikipedia, the free encyclopedia
In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. In particular, the homotopy must go through immersions and extend continuously to a homotopy of the tangent bundle.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them.
The Whitney-Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if their Gauss maps have the same winding number. Stephen Smale classified the regular homotopy classes of an k-sphere immersed in 
. A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in 
. In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".
[edit] References
- Hassler Whitney, On regular closed curves in the plane. Compositio Mathematica, 4 (1937), p. 276-284
- Stephen Smale, A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281--290.
- Stephen Smale, The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69 1959 327--344.
