Immersion

Latest revision as of 14:15, 16 May 2013

1 Definition

We work in a fixed category CAT of topological, piecewise linear, ${{Authors|Ulrich Koschorke}}{{Stub}} ==Definition== <wikitex>; 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 {\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. #''Existence'': Given $M$ and $N$, is there any immersion $ M \looparrowright N $ at all? #''Classification'': How many ''`essentially different´'' immersions exist? </wikitex> ==The smooth case:CAT = DIFF== <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 $\infty$. {{beginthm|Theorem|(Smale--Hirsch 1959/Phillips 1967)}}\label{thm:2.1} 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} 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}} {{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<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}} </wikitex> ==Selfintersections== <wikitex>; 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--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 $(1 \leq r \leq \infty )$ 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 $m \leq n$.

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 Boy's surface, the image of a $C^{\infty}$-immersion: see for example the page on surfaces. 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

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 2.1 [Hirsch1959], [Smale1959a], 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.

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

3 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 $f: M^n \looparrowright \R^{n+1}$.

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

• [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 $n$-manifold in $(2n-1)$-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 $f \colon M^m \rightarrow N^n$ be such a map between manifolds of the indicated dimensions $m \leq n$.

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 Boy's surface, the image of a $C^{\infty}$-immersion: see for example the page on surfaces. 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

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 2.1 [Hirsch1959], [Smale1959a], 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.

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

3 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 $f: M^n \looparrowright \R^{n+1}$.

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

• [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 $n$-manifold in $(2n-1)$-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