# Immersion

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## Contents |

## 1 Definition

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 differentÂ´*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 *mono*morphism . 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 .

**Theorem 2.1** [Hirsch1959], [Smale1959a], Phillips 1967**.**
If (i) , or if (ii) is open and , then the map

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.

**Remark 2.2.**
For a good exposition of Theorem 2.1 see [Adachi1993, pp.87 and 93].

**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.

## 4 References

- [Adachi1993] M. Adachi,
*Embeddings and immersions*, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001 - [Eccles1981] P. J. Eccles,
*Codimension one immersions and the Kervaire invariant one problem*, Math. Proc. Cambridge Philos. Soc.**90**(1981), no.3, 483–493. MR628831 (83c:57015) Zbl 0479.57016 - [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel,
*On the non-existence of elements of Kervaire invariant one*, (2009). Available at the arXiv:0908.3724. - [Hirsch1959] M. W. Hirsch,
*Immersions of manifolds*, Trans. Amer. Math. Soc.**93**(1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603 - [Koschorke1981] U. Koschorke,
*Vector fields and other vector bundle morphisms-a singularity approach*, Springer, 1981. MR611333 (82i:57026) Zbl 0459.57016 - [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff,
*Characteristic classes*, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504 - [Smale1959a] S. Smale,
*The classification of immersions of spheres in Euclidean spaces*, Ann. of Math. (2)**69**(1959), 327–344. MR0105117 (21 #3862) Zbl 0089.18201 - [Smale1963] S. Smale,
*A survey of some recent developments in differential topology*, Bull. Amer. Math. Soc.**69**(1963), 131–145. MR0144351 (26 #1896) Zbl 0133.16507 - [Steenrod1951] N. Steenrod,
*The topology of fibre bundles.*, (Princeton Mathematical Series No. 14.) Princeton: Princeton University Press. VIII, 224 p. , 1951. MR1688579 (2000a:55001) Zbl 0054.07103 - [Whitney1944a] H. Whitney,
*The singularities of a smooth -manifold in -space*, Ann. of Math. (2)**45**(1944), 247â€“293. MR0010275 (5,274a) Zbl 0063.08238

## 5 External links

- The Encyclopedia of Mathematics article on an immersion of manifold
- The Wikipedia page about immersions