# Immersion

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## 1 Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$${{Authors|Ulrich Koschorke}} ==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 . {{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: # 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 . We call f an ''immersion'' (and we write f : M \looparrowright N ) if f is a local immersion at every point x \in M. {{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 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]]. The following two questions play an important role. #''Existence'': Given M and N, is there any immersion M \looparrowright N at all? #''Classification'': How many ''essentially different´'' immersions exist? ==The smooth case == ; 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|\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 $$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. {{endthm}} {{beginrem|Remark}} For a good exposition of Theorem \ref{thm:2.1} see {{cite|Adachi1993|pp.87 and 93}}. {{endrem}} {{beginthm|Corollary|}}\label{cor:2.2} Under the assumptions of theorem \ref{thm: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 M to TN . E. g. if M is parallelizable (i. e. TM \cong M \times \R^m ) then M \looparrowright \Rr^{m+1} . {{endthm}} {{beginthm|Theorem|\cite{Whitney1944a} }}\label{thm:2.3} 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). {{endthm}} {{beginrem|Remark}} See also e.g. {{cite|Adachi1993|p. 86ff}}. {{endrem}} Theorem \ref{thm:2.3} is best possible as long as we put no restrictions on M. {{beginrem|Example|}}\label{exa:2.4} 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, \cite{Milnor&Stasheff1974|Theorem 4,8}. {{endrem}} {{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: # f_0 = f, \quad f_1 = g ; # f_t is a immersion for all t \in I . {{endrem}} {{beginthm|Corollary|}}\label{cor:2.6} 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. {{endthm}} {{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 ==Self intersections== ; 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} . {{beginthm|Theorem|\cite{Eccles1981} }}\label{thm:3.1} 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 C^r$-differentiable $(1 \leq r \leq \infty )$$(1 \leq r \leq \infty )$ or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them. $\mathring{B}^k$$\mathring{B}^k$ denotes the open unit ball in $\R^k, k = 0,1, \ldots$$\R^k, k = 0,1, \ldots$.

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

Definition 1.1. $f$$f$ is a local immersion at a point $x \in M$$x \in M$ if there exist open neighbourhoods $U$$U$ of $x$$x$ and $V$$V$ of $f(x)$$f(x)$ in $M$$M$ and $N$$N$, resp., such that $f(U) \subset V$$f(U) \subset V$ and:

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

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

Thus an immersion looks locally like the inclusion $\R^m \subset \R^n$$\R^m \subset \R^n$ of Euclidean spaces. It allows us to visualize a given manifold $M$$M$ in a possibly more familar setting such as $N = \R^n$$N = \R^n$. E.g. the projective plane $\RP^2$$\RP^2$ can be visualized in $\R^3$$\R^3$ with the help of the Boy's surface, the image of a $C^{\infty}$$C^{\infty}$-immersion: see for example the page on surfaces. The following two questions play an important role.

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

## 2 The smooth case

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

Theorem 2.1 [Hirsch1959], [Smale1959a], Phillips 1967. If (i) $m < n$$m < n$, or if (ii) $M$$M$ is open and $m=n$$m=n$, then the map

(1)$T : \textup{Imm}(M,N) \rightarrow \textup{Mono}(TM,TN), \quad f \rightarrow Tf,$$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)$$\textup{Imm}(M, N)$, of all smooth immersions $f : M \looparrowright N$$f : M \looparrowright N$ and $\textup{Mono}(TM, TN)$$\textup{Mono}(TM, TN)$, the space of all vector bundle monomorphisms $\varphi : TM \rightarrow TN$$\varphi : TM \rightarrow TN$, are endowed respectively with the $C^{\infty}$$C^{\infty}$-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 $f: M \looparrowright N$$f: M \looparrowright N$ if and only if there is a vector bundle monomorphism from the tangent bundle $TM$$TM$ of $M$$M$ to $TN$$TN$. E. g. if $M$$M$ is parallelizable (i. e. $TM \cong M \times \R^m$$TM \cong M \times \R^m$) then $M \looparrowright \Rr^{m+1}$$M \looparrowright \Rr^{m+1}$.

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

Theorem 2.4 is best possible as long as we put no restrictions on $M$$M$.

Example 2.6. The real projective space $\RP^m$$\RP^m$ cannot be immersed into $\R^{2m-2}$$\R^{2m-2}$ if $m = 2^k$$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$$f, g : M \looparrowright N$ are regularly homotopic if there exists a smooth map $F : M \times I \rightarrow N$$F : M \times I \rightarrow N$ which with $f_t(x) := F(x,t)$$f_t(x) := F(x,t)$ satisfies the following:

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

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

Example 2.9 $M = S^m, N = \R^n$$M = S^m, N = \R^n$, [Smale1959]. The regular homotopy classes of immersions $f: S^m \looparrowright \R^n, \ m$f: S^m \looparrowright \R^n, \ m, are in one-to-one correspondance with the elements of the homotopy group $\pi_m(V_{n,m})$$\pi_m(V_{n,m})$, where $V_{n,m}$$V_{n,m}$ is the Stiefel manifold of $m$$m$-frames in $\R^n$$\R^n$. In particular, all immersions $S^2 \looparrowright \R^3$$S^2 \looparrowright \R^3$ are regularly homotopic (since $\pi_2(V_{3,2}) = 0$$\pi_2(V_{3,2}) = 0$). E. g. the standard inclusion $f_0 : S^2 \subset \R^3$$f_0 : S^2 \subset \R^3$ is regularly homotopic to $-f_0$$-f_0$; i. e. you can \textit{'turn the sphere inside out'} in $\R^3$$\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 Self intersections

It is a characteristic feature of immersions - as compared to embeddings - that $r$$r$-tuple selfintersections may occur for some $r \geq 2$$r \geq 2$, i. e. points in $N$$N$ which are the image of at least $r$$r$ distinct elements of M (e. g. the double point in the figure 8 immersion $f : S^1 \looparrowright \R^2$$f : S^1 \looparrowright \R^2$ with image $\infty$$\infty$). Generically the locus of r-tuple points of a smooth immersion $f: M^m \looparrowright N^n$$f: M^m \looparrowright N^n$ is an immersed ($n- r(n-m)$$n- r(n-m)$)-dimensional manifold in $N$$N$. Its properties may yield a variety of interesting invariants which link immersions to other concepts of topology. E. g. let $\theta(f)$$\theta(f)$ denote the $\mod{2}$$\mod{2}$ number of ($n+1$$n+1$)-tuple points of a selftransverse immersion $f: M^n \looparrowright \R^{n+1}$$f: M^n \looparrowright \R^{n+1}$.

Theorem 3.1 [Eccles1981] . Given a natural number $n \equiv 1(4)$$n \equiv 1(4)$, there is an $n$$n$-dimensional closed smooth manifold $M^n$$M^n$ and an immersion $f : M^n \looparrowright \R^{n+1}$$f : M^n \looparrowright \R^{n+1}$ satisfying $\ \theta(f) = 1$$\ \theta(f) = 1$ if and only if there exists a framed ($n+1$$n+1$)-dimensional manifold with Kervaire invariant $1$$1$.

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