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. 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 → N {displaystyle f,g:M o N} are homotopic if they represent points in the same path-components of the mapping space C ( M , N ) {displaystyle C(M,N)} , given the compact-open topology. The space of immersions is the subspace of C ( M , N ) {displaystyle C(M,N)} consisting of immersions, denote it by I m m ( M , N ) {displaystyle Imm(M,N)} . Two immersions f , g : M → N {displaystyle f,g:M o N} are regularly homotopic if they represent points in the same path-component of I m m ( M , N ) {displaystyle Imm(M,N)} . 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 R n {displaystyle 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. A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in R 3 {displaystyle 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.