Immersion

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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.

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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 M 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 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 {\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 M 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 well-defined tangent line. However there exists a smooth immersion f : S^1 \looparrowright \Rr^2 with image the figure \infty.

Theorem 1.2 (Smale-Hirsch 1959/Phillips 1967). If (i) m < n, or if (ii) M 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 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.

Remark 1.3. For a good exposition of Theorem 1.2 see [Adachi1993, pp.87 and 93].

Corollary 1.4. Under the assumptions of theorem 1.2 there exists an immersion f: M \looparrowright N if and only if there is a vector bundle monomorphism from the tangent bundle TM of M to TN. E. g. if M is parallelizable (i. e. TM \cong M \times \R^m) then M \looparrowright \Rr^{m+1}.

Theorem 1.5 [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 1.6. See also e.g. [Adachi1993, p. 86ff].

Theorem 1.5 is best possible as long as we put no restrictions on M.

Example 1.7. 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 1.8. 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 1.9. 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 1.10 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 2.1 [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 M 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

5 External links

$. {{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:}} == 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]]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 M 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 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 {\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 M 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 well-defined tangent line. However there exists a smooth immersion f : S^1 \looparrowright \Rr^2 with image the figure \infty.

Theorem 1.2 (Smale-Hirsch 1959/Phillips 1967). If (i) m < n, or if (ii) M 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 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.

Remark 1.3. For a good exposition of Theorem 1.2 see [Adachi1993, pp.87 and 93].

Corollary 1.4. Under the assumptions of theorem 1.2 there exists an immersion f: M \looparrowright N if and only if there is a vector bundle monomorphism from the tangent bundle TM of M to TN. E. g. if M is parallelizable (i. e. TM \cong M \times \R^m) then M \looparrowright \Rr^{m+1}.

Theorem 1.5 [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 1.6. See also e.g. [Adachi1993, p. 86ff].

Theorem 1.5 is best possible as long as we put no restrictions on M.

Example 1.7. 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 1.8. 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 1.9. 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 1.10 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 2.1 [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 M 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

5 External links

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