Immersion

(Difference between revisions)
Jump to: navigation, search
(→The smooth case:CAT = DIFF)
(→The smooth case)
Line 19: Line 19:
</wikitex>
</wikitex>
−
==The smooth case==
+
==The smooth case: CAT = DIFF==
<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 welldefined tangent line. However there exists a smooth immersion $f : S^1 \looparrowright \Rr^2 $ with image the figure $\infty$.
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 welldefined tangent line. However there exists a smooth immersion $f : S^1 \looparrowright \Rr^2 $ with image the figure $\infty$.
Line 26: Line 26:
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 sets of all smooth immersions $f : M \looparrowright N $ and of all vector bundle monomorphisms $ \varphi : TM \rightarrow TN $, resp., are endowed with the $ C^{\infty}$--topology and the compact--open topology, resp.
+
is a weak homotopy equivalence. Here the sets of all smooth immersions $f : M \looparrowright N $ and of all vector bundle monomorphisms $ \varphi : TM \rightarrow TN $, resp., are endowed with the $ C^{\infty}$-topology and the compact-open topology, resp.
{{endthm}}
{{endthm}}
Line 62: Line 62:
{{beginrem|Example|$ M = S^m, N = \R^n$, \cite{Smale1959}}}\label{exa:2.7}
{{beginrem|Example|$ M = S^m, N = \R^n$, \cite{Smale1959}}}\label{exa:2.7}
−
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 \textit{'turn the sphere inside out'} in $\R^3$, with possible selfintersections but without creating any crease. 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.
+
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 \textit{'turn the sphere inside out'} in $\R^3$, with possible selfintersections but without creating any crease. 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.
{{endrem}}
{{endrem}}
</wikitex>
</wikitex>

Revision as of 16:24, 4 April 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.

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Definition

We work in a fixed category CAT of topological, piecewise linear, C^r-differentiable (1 \leq r \leq \infty ) or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them. \mathring{B}^k denotes the open unit ball in \R^k, k = 0,1, \ldots.

Let f \colon M^m \rightarrow N^n be such a map between manifolds of the indicated dimensions m \leq n.

Definition 1.1.

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 
Tex syntax error
 and N, resp., such that f(U) \subset V and:
  1. 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
  2. h \circ f yields a CAT-isomorphism from U onto \mathring{B}^m.

We call f an immersion (and we write f : M \looparrowright N) if f is a local immersion at every point x \in M.

Thus an immersion looks locally like the inclusion \R^m \subset \R^n of Euclidean spaces. It allows us to visualize a given manifold 
Tex syntax error
 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 {\itshape Boy's surface}, the image of a C^{\infty}--immersion (beautiful illustrations can be found in the internet).

The following two questions play an important role.

  1. Existence: Given 
    Tex syntax error
     and N, is there any immersion M \looparrowright N at all?
  2. Classification: How many `essentially different´ immersions exist?

2 The smooth case: CAT = DIFF

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 monomorphism 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 welldefined tangent line. However there exists a smooth immersion f : S^1 \looparrowright \Rr^2 with image the figure \infty.

Theorem 2.1 (Smale-Hirsch 1959/Phillips 1967).

If (i) m < n, or if (ii) 
Tex syntax error
 is open and m=n, then the map
(1)T : \textup{Imm}(M,N) \rightarrow \textup{Mono}(TM,TN), \quad f \rightarrow Tf,

is a weak homotopy equivalence. Here the sets of all smooth immersions f : M \looparrowright N and of all vector bundle monomorphisms \varphi : TM \rightarrow TN, resp., are endowed with the C^{\infty}-topology and the compact-open topology, resp.

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 f: M \looparrowright N if and only if there is a vector bundle monomorphism from the tangent bundle TM of 
Tex syntax error
 to TN. E. g. if 
Tex syntax error
 is parallelizable (i. e. TM \cong M \times \R^m) then M \looparrowright \Rr^{m+1}.

Theorem 2.4 [Whitney1944a] . If m \geq 2 then there exists an immersion M^m \looparrowright \R^{2m-1} (E. g. any surface can be immersed into \R^3).

Remark 2.5. See also e.g. [Adachi1993, p. 86ff].

Theorem 2.4 is best possible as long as we put no restrictions on 
Tex syntax error
.

Example 2.6. The real projective space \RP^m cannot be immersed into \R^{2m-2} if m = 2^k. This follows from an easy calculation using Stiefel-Whitney classes: see, [Milnor&Stasheff1974, Theorem 4,8].

Definition 2.7. 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:

  1. f_0 = f, \quad f_1 = g;
  2. f_t is a immersion for all t \in I.

Corollary 2.8. Assume m < n. Two immersions f, g : M \looparrowright N are regularly homotopic if and only if their tangent maps Tf, Tg : TM \rightarrow TN are homotopic through vector bundle monomorphisms.

Example 2.9 M = S^m, N = \R^n, [Smale1959]. 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 \textit{'turn the sphere inside out'} in \R^3, with possible selfintersections but without creating any crease. 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.

3 Selfintersections

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 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}.

Theorem 3.1 [Eccles1981] . 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.

According to [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 
Tex syntax error
 in question cannot be orientable (cf. [Eccles1981]). Thus the figure 8 immersion f : S^1 \looparrowright \R^2 plays a rather special role here.

4 References

$. {{endthm}} According to \cite{Hill&Hopkins&Ravenel2009} (and previous authors) this holds precisely when $ n=1, 5, 13, 29, 61 $ or possibly 5$. 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. == References == {{#RefList:}} [[Category:Definitions]]C^r-differentiable (1 \leq r \leq \infty ) or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them. \mathring{B}^k denotes the open unit ball in \R^k, k = 0,1, \ldots.

Let f \colon M^m \rightarrow N^n be such a map between manifolds of the indicated dimensions m \leq n.

Definition 1.1.

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 
Tex syntax error
 and N, resp., such that f(U) \subset V and:
  1. 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
  2. h \circ f yields a CAT-isomorphism from U onto \mathring{B}^m.

We call f an immersion (and we write f : M \looparrowright N) if f is a local immersion at every point x \in M.

Thus an immersion looks locally like the inclusion \R^m \subset \R^n of Euclidean spaces. It allows us to visualize a given manifold 
Tex syntax error
 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 {\itshape Boy's surface}, the image of a C^{\infty}--immersion (beautiful illustrations can be found in the internet).

The following two questions play an important role.

  1. Existence: Given 
    Tex syntax error
     and N, is there any immersion M \looparrowright N at all?
  2. Classification: How many `essentially different´ immersions exist?

2 The smooth case: CAT = DIFF

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 monomorphism 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 welldefined tangent line. However there exists a smooth immersion f : S^1 \looparrowright \Rr^2 with image the figure \infty.

Theorem 2.1 (Smale-Hirsch 1959/Phillips 1967).

If (i) m < n, or if (ii) 
Tex syntax error
 is open and m=n, then the map
(1)T : \textup{Imm}(M,N) \rightarrow \textup{Mono}(TM,TN), \quad f \rightarrow Tf,

is a weak homotopy equivalence. Here the sets of all smooth immersions f : M \looparrowright N and of all vector bundle monomorphisms \varphi : TM \rightarrow TN, resp., are endowed with the C^{\infty}-topology and the compact-open topology, resp.

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 f: M \looparrowright N if and only if there is a vector bundle monomorphism from the tangent bundle TM of 
Tex syntax error
 to TN. E. g. if 
Tex syntax error
 is parallelizable (i. e. TM \cong M \times \R^m) then M \looparrowright \Rr^{m+1}.

Theorem 2.4 [Whitney1944a] . If m \geq 2 then there exists an immersion M^m \looparrowright \R^{2m-1} (E. g. any surface can be immersed into \R^3).

Remark 2.5. See also e.g. [Adachi1993, p. 86ff].

Theorem 2.4 is best possible as long as we put no restrictions on 
Tex syntax error
.

Example 2.6. The real projective space \RP^m cannot be immersed into \R^{2m-2} if m = 2^k. This follows from an easy calculation using Stiefel-Whitney classes: see, [Milnor&Stasheff1974, Theorem 4,8].

Definition 2.7. 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:

  1. f_0 = f, \quad f_1 = g;
  2. f_t is a immersion for all t \in I.

Corollary 2.8. Assume m < n. Two immersions f, g : M \looparrowright N are regularly homotopic if and only if their tangent maps Tf, Tg : TM \rightarrow TN are homotopic through vector bundle monomorphisms.

Example 2.9 M = S^m, N = \R^n, [Smale1959]. 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 \textit{'turn the sphere inside out'} in \R^3, with possible selfintersections but without creating any crease. 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.

3 Selfintersections

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 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}.

Theorem 3.1 [Eccles1981] . 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.

According to [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 
Tex syntax error
 in question cannot be orientable (cf. [Eccles1981]). Thus the figure 8 immersion f : S^1 \looparrowright \R^2 plays a rather special role here.

4 References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Immersion&oldid=10407"
Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox