Immersion (mathematics)

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In mathematics, an immersion is a differentiable map between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if

is an injective map at every point p of M (where the notation T_pX represents the tangent space of X at the point p). Equivalently, f is an immersion if it has constant rank equal to the dimension of M:

The map f itself need not be injective, only its derivative.

A related concept is that of an embedding. A smooth embedding is an injective immersion f : M → N which is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is a local embedding (i.e. for any point there is a neighbourhood, , of x such that is an embedding.)

Contents 1 Regular homotopy 2 Multiple points 3 Examples and properties 4 See also 5 References

[edit] Regular homotopy

A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H: M × [0,1] → N such for all the function H_t: M → N defined by H_t(x)=H(x,t) for all is an immersion, with H₀=f, H₁=g. A regular homotopy is thus a homotopy through immersions.

Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940's, proving that for 2m < n + 1 every map f : M^m -→ Nⁿ of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n. Stephen Smale expressed the regular homotopy classes of immersions f : M^m -→ Rⁿ as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence. Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any m-dimensional manifold M^m in any n-dimensional manifold Nⁿ. The Hirsch-Smale classification of immersions was generalized by Misha Gromov.

[edit] Multiple points

A k-tuple point of an immersion f : M → N is an unordered set of distinct points with the same image . If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion f : M → N in general position the set of k-tuple points is an n − k(n − m)-dimensional manifold.

A double point is a k-tuple point with k = 2. An embedding is an immersion without double (or higher multiplicity) points, i.e. an embedding is an injective immersion.

The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points.

At a key point in surgery theory it is necessary to decide if an immersion f : S^m -→ N^2m of an m-sphere in a 2m-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery. Wall associated to f an invariant μ(f) in a quotient of the fundamental group ring Z[π₁(N)] which counts the double points of f in the universal cover of N. For m > 2 f is regular homotopic to an embedding if and only if μ(f) = 0 by the Whitney trick.

[edit] Examples and properties

• A mathematical rose with k petals is an immersion of the circle in the plane with a single k-tuple point. For k = 2 this is a figure 8.
• By the Whitney-Graustein theorem the regular homotopy classes of immersions of the circle in the plane are classified by the winding number which is also the number of double points.
• The sphere can be turned inside out: the standard embedding is related to by a regular homotopy of immersions .

[edit] See also

• Submersion
• Immersed submanifold

[edit] References

• Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985), Singularities of Differentiable Maps: Volume 1, Birkhäuser, ISBN 0817631879 
• Bruce, J. W.; Giblin, P. J. (1984), Curves and Singularities, Cambridge University Press, ISBN 0521429994 
• Gromov, M. (1986), Partial differential relations, Springer, ISBN 3-540-12177-3 
• Hirsch M. Immersions of manifolds. Trans. A.M.S. 93 1959 242--276.
• Smale, S. A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281–290.
• Smale, S. The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69 1959 327--344.#
• Wall, C.T.C.: Surgery on compact manifolds. 2nd ed., Mathematical Surveys and Monographs 69, A.M.S.