# Difference between revisions of "Immersion"

A mapping $f: X \rightarrow Y$ of one topological space into another for which each point of $X$ has a neighbourhood $U$ which is homeomorphically mapped onto $fU$ by $f$. This concept is applied mainly to mappings of manifolds, where one often additionally requires a local flatness condition (as for a locally flat imbedding). The latter condition is automatically fulfilled if the manifolds $X$ and $Y$ are differentiable and if the Jacobi matrix of the mapping $f$ has maximum rank, equal to the dimension of $X$ at each point. The classification of immersions of one manifold into another up to a regular homotopy can be reduced to a pure homotopic problem. A homotopy $f _ {t} : X ^ {m} \rightarrow Y ^ {n}$ is called regular if for each point $x \in X$ it can be continued to an isotopy (in topology) $F _ {t} : U \times D ^ {k} \rightarrow Y$, where $U$ is a neighbourhood of $x$, $D ^ {k}$ is a disc of dimension $k = n- m$ and $F _ {t}$ coincides with $f _ {t}$ on $U \times 0$, where 0 is the centre of the disc. In the differentiable case, it is sufficient to require that the Jacobi matrix has maximum rank for each $t$ and depends continuously on $t$. The differential $D _ {f}$ of an immersion determines a fibre-wise monomorphism of the tangent bundle $\tau X$ into the tangent bundle $\tau Y$. A regular homotopy determines a homotopy of such monomorphisms. This establishes a bijection between the classes of regular homotopies and the homotopy classes of monomorphisms of bundles.
The problem of immersions in a Euclidean space reduces to the homotopy classification of mappings into a Stiefel manifold $V _ {n,m }$. For example, because $\pi _ {2} ( V _ {3,2 } ) = 0$, there is only one immersion class of the sphere $S ^ {2}$ into $\mathbf R ^ {3}$, so the standard imbedding is regularly homotopic to its mirror reflection (the sphere may be regularly turned inside out. Because $V _ {2,1 } \approx S ^ {1}$, there is a countable number of immersion classes of a circle into the plane, and because the Stiefel fibration over $S ^ {2}$ is homeomorphic to the projective space $\mathbf R P ^ {3}$ and $\pi _ {1} ( \mathbf R P ^ {3} ) = \mathbf Z _ {2}$, there are only two immersion classes from $S ^ {1}$ into $S ^ {2}$, etc.
For figures illustrating the fact that $S ^ {2}$ can be regularly turned inside out see [a3].