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In the [[mathematics|mathematical]] field of [[topology]], a '''regular homotopy''' refers to a special kind of [[homotopy]] between [[immersion (mathematics)|immersion]]s of one [[manifold]] in another. The homotopy must be a 1-parameter family of immersions. | |||
Similar to [[homotopy class]]es, 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 [[Homotopy#Isotopy|isotopy]] of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions <math>f,g : M \to N</math> are homotopic if they represent points in the same path-components of the mapping space <math>C(M,N)</math>, given the [[compact-open topology]]. The '''space of immersions''' is the subspace of <math>C(M,N)</math> consisting of immersions, denote it by <math>Imm(M,N)</math>. Two immersions <math>f,g:M \to N</math> are '''regularly homotopic''' if they represent points in the same path-component of <math>Imm(M,N)</math>. | |||
== Examples == | |||
[[File:Winding Number Around Point.svg|thumb|300px|This curve has [[total curvature]] 6''π'', and [[turning number]] 3.]] | |||
The '''Whitney–Graustein theorem''' {{anchor|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 map]]s have the same degree/[[winding number]]. | |||
[[File:MorinSurfaceFromTheTop.PNG|thumb|Smale's classification of immersions of spheres shows that [[sphere eversion]]s exist, which can be realized via this [[Morin surface]].]] | |||
[[Stephen Smale]] classified the regular homotopy classes of a ''k''-sphere immersed in <math>\mathbb R^n</math> – they are classified by homotopy groups of [[Stiefel manifold]]s, which is a generalization of the Gauss map, with here ''k'' partial derivatives not vanishing. A corollary of his work is that there is only one regular homotopy class of a ''2''-sphere immersed in <math>\mathbb R^3</math>. In particular, this means that [[sphere eversion]]s 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== | |||
*[[Hassler Whitney]], ''[http://www.numdam.org/numdam-bin/item?id=CM_1937__4__276_0 On regular closed curves in the plane]''. [[Compositio Mathematica]], 4 (1937), p. 276–284 | |||
*Stephen Smale, ''A classification of immersions of the two-sphere''. [[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]] 90 1958 281–290. | |||
*Stephen Smale, ''The classification of immersions of spheres in Euclidean spaces''. [[Annals of Mathematics|Ann. of Math.]] (2) 69 1959 327–344. | |||
[[Category:Differential topology]] | |||
[[Category:Algebraic topology]] | |||
Revision as of 07:08, 9 October 2013
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 are homotopic if they represent points in the same path-components of the mapping space , given the compact-open topology. The space of immersions is the subspace of consisting of immersions, denote it by . Two immersions are regularly homotopic if they represent points in the same path-component of .
Examples
The Whitney–Graustein theorem <Whitney–Graustein theorem>...</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 – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing. 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".
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
- 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.