MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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:… ,… 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
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.

share|cite|improve this answer

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?

share|cite|improve this answer
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
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 $F|_{\partial N}=f$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.