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.

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 ${\displaystyle f,g:M\to N}$ are homotopic if they represent points in the same path-components of the mapping space ${\displaystyle C(M,N)}$, given the compact-open topology. The space of immersions is the subspace of ${\displaystyle C(M,N)}$ consisting of immersions, denoted by ${\displaystyle \operatorname {Imm} (M,N)}$. Two immersions ${\displaystyle f,g:M\to N}$ are regularly homotopic if they represent points in the same path-component of ${\displaystyle \operatorname {Imm} (M,N)}$.