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Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?

Remark: If the sphere had dimension k smaller than n-1, then such an immersion would exist if and only if the corresponding map from the k-sphere to the Stiefel manifold is 0-homotopic. This is the Hirsch-Smale Theorem and in fact an example of an h-principle. However the case k=n-1 is exactly the exceptional case which does NOT obey an h-principle. Easy examples (Figure 8.1. in the book by Eliashberg-Mishachev) show that there exist immersions of the circle in the plane which have a formal extension but not a genuine extension to the 2-disk. So, is there anything known about sufficient conditions for extendability?

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I'm probably ignorant, but can you say why this question is tagged ds.dynamical-systems? –  Willie Wong Jun 16 '11 at 16:02
    
Related: mathoverflow.net/questions/57215/… , mathoverflow.net/questions/43743/… and the work of Koschorke on singularities of bundle morphisms. But I think the question is a subtle one. –  Mark Grant Jun 16 '11 at 16:14
    
@MG: Wasn't Koschorke's work about codimension one immersions, which then reduces to the study of vector bundle monomorphisms? –  ThiKu Jun 16 '11 at 16:43
    
@unknown (google): Koschorke's work is quite general, but I'm not sure it applies to this problem. See Chapter 1.3 of "Vector fields and other vector bundle morphisms—a singularity approach", where a complete obstruction to a map being homotopic to an immersion is constructed. It is an element in a certain normal bordism group. –  Mark Grant Jun 17 '11 at 16:02

3 Answers 3

up vote 14 down vote accepted

This is subtle, even for $n=2$. In this case, clearly the problem reduces to $S^2$ or $\mathbb{R}^2$ since every surface has one of these as a universal cover. Samuel Blank found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposition has been given by Valentin Poenaru, and the criterion has been extended to $S^2$ by Frisch. There is also a bit of discussion in these papers about the higher dimensional problem.

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Smale-Hirsch is not just a theorem about existence of immersions. It's a theorem about the homotopy-type of the space of all immersions.

Given an immersion $$S^{n-1} \to \mathbb R^n$$

you get a bundle monomorphism

$$TS^{n-1} \to \mathbb R^n$$

There's a cute trick that shows the space of all such bundle monomorphisms has the homotopy-type of $Maps(S^{n-1}, SO_n)$. Here's how it goes. Given a bundle monomorphism $f : TS^{n-1} \to \mathbb R^n$ the associated map $G(f) : S^{n-1} \to SO_n$ is defined by, given $p \in S^{n-1}$ and $v \in \mathbb R^n$. Then $G(f)(p)(v)$ is defined by letting $v_\perp \in \mathbb R$ and $V_{||} \in T_pS^{n-1}$ be the orthogonal component and tangent-space orthogonal projection of $v$, and $G(f)(p)(v) = f(p)(v_{||}) + v_{\perp}f(p)^+$ where $f(p)^+$ is the unit vector normal to $f(p)(T_pS^{n-1})$ chosen so that $G(f) \in SO_n$ i.e. that it is not orientation-reversing. You can reverse this construction as well, to go from maps $S^{n-1} \to SO_n$ to bundle immersions $TS^{n-1} \to \mathbb R^n$.

It's basically by design, a homotopy of $G(f)$ can be re-interpreted as a $1$-parameter family of immersions $S^{n-1} \to \mathbb R^n$ equipped with a normal vector field.

Perhaps you can't extend this 1-parameter family to an immersion $S^{n-1} \times [0,1] \to \mathbb R^n$. Is that the key issue?

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I'm not so sure whether this works. The problem might be that a homotopy of immersions of S does not necessarily yield an immersion of Sx[0,1]. One needs that the derivative in [0,1]-direction is linearly independent of the derivatives in the S-direction. –  ThiKu Jun 16 '11 at 23:46
    
There's a key difference. A map $X \to V_{n,j}$ may not lift to a map $X \to V_{n,j+1}$. But a map $X \to V_{n,n-1} \equiv SO_n$ always lifts to a map $X \to V_{n,n} \equiv O_n$. Let me edit my answer a bit to make the key step less "insiderish". –  Ryan Budney Jun 17 '11 at 0:23
    
I don't think constructing the 1-parameter family of immersions with a normal vector field is the problem. But I changed my answer. It's only a partial response to your question, not really everything you were looking for. What do you mean by "formal extension" -- is that the 1-parameter family of immersions with normal vector field? –  Ryan Budney Jun 17 '11 at 1:13
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Smale-Hirsch describes the homotopy types of these spaces of immersions, as Ryan says, so it gives a test for whether a given immersion of $S^{n-1}$ in an $n$-manifold is homotopic through immersions to one that extends to an immersion of $D^n$. But it does not answer the question of whether a given immersion can be so extended. You might think that the restriction map from the space of immersions of $D^n$ to the space of immersions of $S^{n-1}$ is a fibration, but it's not. It is if the disk has positive codimension, and this is a key step in proving Smale-Hirsch. But it's false in codim $0$. –  Tom Goodwillie Jun 17 '11 at 2:07
    
@ RB : "formal immersion" means a vector bundle monomorphism TM--->TN which does not necessarily come from an immersion M--->N. If dim(M)<dim(N), then every formal immersion is homotopic to an immersion, but for dim(M)=dim(N) this is not always true. –  ThiKu Jun 18 '11 at 11:26

Christian Pappas gave a Morse-theoretic method for constructing all extensions of a codimension 1 immersion $f:\partial N\to W$ to an immersion $F:N\to W$ with $F|_{\partial N}=f$.

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