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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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Regular homotopy」の詳細全文を読む スポンサード リンク
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