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We work in a fixed category CAT of topological, piecewise linear, -differentiable or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them. denotes the open unit ball in .
Let be such a map between manifolds of the indicated dimensions .
Definition 1.1. is a local immersion at a point if there exist open neighbourhoods of and of in and , resp., such that and:
- there is a CAT-isomorphism (i.e. both and are CAT-maps) which maps onto ; and
- yields a CAT-isomorphism from onto .
We call f an immersion (and we write ) if is a local immersion at every point .
Thus an immersion looks locally like the inclusion of Euclidean spaces. It allows us to visualize a given manifold in a possibly more familar setting such as . E.g. the projective plane can be visualized in with the help of the Boy's surface, the image of a -immersion: see for example the page on surfaces. The following two questions play an important role.
- Existence: Given and , is there any immersion at all?
- Classification: How many `essentially different´ immersions exist?
2 The smooth case
This section is about the category of smooth, i.e. , manifolds and maps. It follows from the inverse function theorem that a smooth map between smooth manifolds is a local immersion at precisely if the tangent map is injective. Thus is a smooth immersion if and only if it induces a vector bundle monomorphism . E.g. the figure cannot be the image of a smooth immersion, due to the two sharp corners which don't allow a well-defined tangent line. However there exists a smooth immersion with image the figure .
is a weak homotopy equivalence. Here the space, , of all smooth immersions and , the space of all vector bundle monomorphisms , are endowed respectively with the -topology and the compact-open topology.
Corollary 2.3. Under the assumptions of theorem 2.1 there exists an immersion if and only if there is a vector bundle monomorphism from the tangent bundle of to . E. g. if is parallelizable (i. e. ) then .
Theorem 2.4 [Whitney1944a] . If then there exists an immersion (E. g. any surface can be immersed into ).
Remark 2.5. See also e.g. [Adachi1993, p. 86ff].
Theorem 2.4 is best possible as long as we put no restrictions on .
Example 2.6. The real projective space cannot be immersed into if . This follows from an easy calculation using Stiefel-Whitney classes: see, [Milnor&Stasheff1974, Theorem 4,8].
Definition 2.7. Two immersions are regularly homotopic if there exists a smooth map which with satisfies the following:
- is a immersion for all .
Corollary 2.8. Assume . Two immersions are regularly homotopic if and only if their tangent maps are homotopic through vector bundle monomorphisms.
Example 2.9 , [Smale1959a]. The regular homotopy classes of immersions , are in one-to-one correspondance with the elements of the homotopy group , where is the Stiefel manifold of -frames in . In particular, all immersions are regularly homotopic (since ). E. g. the standard inclusion is regularly homotopic to ; i. e. you can turn the sphere inside out in , with possible self-intersections but without creating any crease.
Remark 2.10. The Smale-Hirsch theorem makes existence and classification problems accessible to standard methods of algebraic topology such as classical obstruction theorem (cf. e.g. [Steenrod1951]), characteristic classes (cf. e.g. [Milnor&Stasheff1974]), Postnikov towers, the singularity method (cf. e.g. [Koschorke1981]) etc.: see [Smale1963] for the state of the art in 1963.
3 Self intersections
It is a characteristic feature of immersions - as compared to embeddings - that -tuple selfintersections may occur for some , i. e. points in which are the image of at least distinct elements of M (e. g. the double point in the figure 8 immersion with image ). Generically the locus of r-tuple points of a smooth immersion is an immersed ()-dimensional manifold in . Its properties may yield a variety of interesting invariants which link immersions to other concepts of topology. E. g. let denote the number of ()-tuple points of a selftransverse immersion .
Theorem 3.1 [Eccles1981] . Given a natural number , there is an -dimensional closed smooth manifold and an immersion satisfying if and only if there exists a framed ()-dimensional manifold with Kervaire invariant .
According to [Hill&Hopkins&Ravenel2009] (and previous authors) this holds precisely when or possibly . If and the manifold in question cannot be orientable (cf. [Eccles1981]). Thus the figure 8 immersion plays a rather special role here.
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