# Immersion

(18 intermediate revisions by one user not shown) | |||

Line 1: | Line 1: | ||

− | {{Authors|Ulrich Koschorke | + | {{Authors|Ulrich Koschorke}} |

==Definition== | ==Definition== | ||

<wikitex>; | <wikitex>; | ||

Line 7: | Line 7: | ||

{{beginthm|Definition}} | {{beginthm|Definition}} | ||

$ f $ is a ''local immersion at a point'' $x \in M$ if there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$ in $M$ and $N$, resp., such that $f(U) \subset V $ and: | $ f $ is a ''local immersion at a point'' $x \in M$ if there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$ in $M$ and $N$, resp., such that $f(U) \subset V $ and: | ||

− | # there is a CAT | + | # there is a CAT-isomorphism $h : V \rightarrow \mathring{B}^n $ (i.e. both $h$ and $h^{-1}$ are CAT-maps) which maps $f(U)$ onto $\mathring{B}^n \cap (\R^m \times \lbrace 0 \rbrace) = \mathring{B}^m$; and |

# $h \circ f $ yields a CAT-isomorphism from $U$ onto $\mathring{B}^m $. | # $h \circ f $ yields a CAT-isomorphism from $U$ onto $\mathring{B}^m $. | ||

Line 13: | Line 13: | ||

{{endthm}} | {{endthm}} | ||

− | Thus an immersion looks locally like the inclusion $\R^m \subset \R^n $ of Euclidean spaces. It allows us to visualize a given manifold $M$ in a possibly more familar setting such as $N = \R^n $. E.g. the projective plane $\RP^2 $ can be visualized in $\R^3 $ with the help of | + | Thus an immersion looks locally like the inclusion $\R^m \subset \R^n $ of Euclidean spaces. It allows us to visualize a given manifold $M$ in a possibly more familar setting such as $N = \R^n $. E.g. the projective plane $\RP^2 $ can be visualized in $\R^3 $ with the help of the [[Wikipedia:Boy's suface|''Boy's surface'']], the image of a $C^{\infty}$-immersion: see for example the page on [[2-manifolds#Non-orientable surfaces|surfaces]].<!-- (beautiful illustrations can be found in the internet). --> |

The following two questions play an important role. | The following two questions play an important role. | ||

#''Existence'': Given $M$ and $N$, is there any immersion $ M \looparrowright N $ at all? | #''Existence'': Given $M$ and $N$, is there any immersion $ M \looparrowright N $ at all? | ||

Line 19: | Line 19: | ||

</wikitex> | </wikitex> | ||

− | ==The smooth case | + | ==The smooth case == |

<wikitex>; | <wikitex>; | ||

− | This section is about the category of smooth, i.e. $C^{\infty}$, manifolds and maps. It follows from the inverse function theorem that a smooth map $f : M \rightarrow N $ between smooth manifolds is a local immersion at $ x \in M$ precisely if the tangent map $(Tf)_x : T_xM \rightarrow T_{f(x)}(N) $ is injective. Thus $f$ is a smooth immersion if and only if it induces a vector bundle ''mono''morphism $Tf : TM \rightarrow TN $. E.g. the figure $\heartsuit$ cannot be the image of a smooth immersion, due to the two sharp corners which don't allow a | + | This section is about the category of smooth, i.e. $C^{\infty}$, manifolds and maps. It follows from the inverse function theorem that a smooth map $f : M \rightarrow N $ between smooth manifolds is a local immersion at $ x \in M$ precisely if the tangent map $(Tf)_x : T_xM \rightarrow T_{f(x)}(N) $ is injective. Thus $f$ is a smooth immersion if and only if it induces a vector bundle ''mono''morphism $Tf : TM \rightarrow TN $. E.g. the figure $\heartsuit$ 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 $f : S^1 \looparrowright \Rr^2 $ with image the figure $\infty$. |

− | {{beginthm|Theorem| | + | {{beginthm|Theorem|\cite{Hirsch1959}, \cite{Smale1959a}, Phillips 1967}}\label{thm:2.1} |

If (i) $m < n$, or if (ii) $M$ is open and $m=n$, then the map | If (i) $m < n$, or if (ii) $M$ is open and $m=n$, then the map | ||

\begin{equation} T : \textup{Imm}(M,N) \rightarrow \textup{Mono}(TM,TN), \quad f \rightarrow Tf, \end{equation} | \begin{equation} T : \textup{Imm}(M,N) \rightarrow \textup{Mono}(TM,TN), \quad f \rightarrow Tf, \end{equation} | ||

− | is a weak homotopy equivalence. Here the | + | is a weak homotopy equivalence. Here the space, $\textup{Imm}(M, N)$, of all smooth immersions $f : M \looparrowright N $ and $\textup{Mono}(TM, TN)$, the space of all vector bundle monomorphisms $ \varphi : TM \rightarrow TN $, are endowed respectively with the $ C^{\infty}$-topology and the compact-open topology. |

{{endthm}} | {{endthm}} | ||

Line 52: | Line 52: | ||

{{beginrem|Definition|}}\label{defn:2.5} | {{beginrem|Definition|}}\label{defn:2.5} | ||

− | Two immersions $f, g : M \looparrowright N $ are ''regularly homotopic'' if there exists a smooth map $F : M \times I \rightarrow N $ which with $f_t(x) := F(x,t) $ satisfies the following: | + | Two immersions $f, g : M \looparrowright N $ are ''regularly homotopic'' if there exists a ''smooth map'' $F : M \times I \rightarrow N $ which with $f_t(x) := F(x,t) $ satisfies the following: |

# $f_0 = f, \quad f_1 = g $; | # $f_0 = f, \quad f_1 = g $; | ||

# $f_t$ is a immersion for all $t \in I $. | # $f_t$ is a immersion for all $t \in I $. | ||

Line 61: | Line 61: | ||

{{endthm}} | {{endthm}} | ||

− | {{beginrem|Example|$ M = S^m, N = \R^n$, \cite{ | + | {{beginrem|Example|$M = S^m, N = \R^n$, \cite{Smale1959a}}}\label{exa:2.7} |

− | The regular homotopy classes of immersions $f: S^m \looparrowright \R^n, \ m<n $, are in one | + | The regular homotopy classes of immersions $f: S^m \looparrowright \R^n, \ m<n $, are in one-to-one correspondance with the elements of the homotopy group $\pi_m(V_{n,m})$, where $V_{n,m}$ is the Stiefel manifold of $m$-frames in $\R^n$. In particular, all immersions $ S^2 \looparrowright \R^3 $ are regularly homotopic (since $\pi_2(V_{3,2}) = 0 $). E. g. the standard inclusion $ f_0 : S^2 \subset \R^3 $ is regularly homotopic to $ -f_0$; i. e. you can ''turn the sphere inside out'' in $\R^3$, with possible self-intersections but without creating any crease. |

+ | {{endrem}} | ||

+ | {{beginrem|Remark}} | ||

+ | The Smale-Hirsch theorem makes existence and classification problems accessible to standard methods of algebraic topology such as classical obstruction theorem (cf. e.g. {{cite|Steenrod1951}}), characteristic classes (cf. e.g. {{cite|Milnor&Stasheff1974}}), Postnikov towers, the singularity method (cf. e.g. {{cite|Koschorke1981}}) etc.: see \cite{Smale1963} for the state of the art in 1963. | ||

{{endrem}} | {{endrem}} | ||

</wikitex> | </wikitex> | ||

− | == | + | ==Self intersections== |

<wikitex>; | <wikitex>; | ||

− | It is a characteristic feature of immersions | + | It is a characteristic feature of immersions - as compared to embeddings - that $r$-tuple selfintersections may occur for some $r \geq 2$, i. e. points in $N$ which are the image of at least $r$ distinct elements of M (e. g. the double point in the figure 8<!-- FIXME \ref{} --> |

− | immersion $ f : S^1 \looparrowright \R^2$ with image $\infty$). Generically the locus of r | + | immersion $ f : S^1 \looparrowright \R^2$ with image $\infty$). Generically the locus of r-tuple points of a smooth immersion $ f: M^m \looparrowright N^n $ is an immersed ($ n- r(n-m) $)-dimensional manifold in $N$. Its properties may yield a variety of interesting invariants which link immersions to other concepts of topology. E. g. let $\theta(f) $ denote the $\mod{2} $ number of ($n+1$)-tuple points of a selftransverse immersion $ f: M^n \looparrowright \R^{n+1} $. |

{{beginthm|Theorem|\cite{Eccles1981} }}\label{thm:3.1} | {{beginthm|Theorem|\cite{Eccles1981} }}\label{thm:3.1} | ||

− | Given a natural number $ n \equiv 1(4) $, there is an $n$ | + | Given a natural number $ n \equiv 1(4) $, there is an $n$-dimensional closed smooth manifold $M^n$ and an immersion $ f : M^n \looparrowright \R^{n+1} $ satisfying $\ \theta(f) = 1 $ if and only if there exists a framed ($n+1$)-dimensional manifold with Kervaire invariant $1$. |

{{endthm}} | {{endthm}} | ||

According to \cite{Hill&Hopkins&Ravenel2009} (and previous authors) this holds precisely when $ n=1, 5, 13, 29, 61 $ or possibly $125$. If $ n \neq 1 $ and $n = 1(4)$ the manifold $M$ in question cannot be orientable (cf. \cite{Eccles1981}). Thus the figure 8 immersion $ f : S^1 \looparrowright \R^2 $ plays a rather special role here. | According to \cite{Hill&Hopkins&Ravenel2009} (and previous authors) this holds precisely when $ n=1, 5, 13, 29, 61 $ or possibly $125$. If $ n \neq 1 $ and $n = 1(4)$ the manifold $M$ in question cannot be orientable (cf. \cite{Eccles1981}). Thus the figure 8 immersion $ f : S^1 \looparrowright \R^2 $ plays a rather special role here. | ||

</wikitex> | </wikitex> | ||

+ | |||

== References == | == References == | ||

{{#RefList:}} | {{#RefList:}} | ||

+ | == External links == | ||

+ | * The Encyclopedia of Mathematics article on an [http://www.encyclopediaofmath.org/index.php/Immersion_of_a_manifold immersion of manifold] | ||

+ | * The Wikipedia page about [[Wikipedia:Immersion_(mathematics)|immersions]] | ||

[[Category:Definitions]] | [[Category:Definitions]] |

## Latest revision as of 14:15, 16 May 2013

An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:15, 16 May 2013 and the changes since publication. |

The user responsible for this page is Ulrich Koschorke. No other user may edit this page at present. |

## 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 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 *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), 247–293. MR0010275 (5,274a) Zbl 0063.08238

## 5 External links

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

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 *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), 247–293. MR0010275 (5,274a) Zbl 0063.08238

## 5 External links

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