Codimension zero immersions - MathOverflow

Codimension zero immersions

2011-06-16T15:43:49Z

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?

Answer by Ryan Budney for Codimension zero immersions

2011-06-16T17:48:21Z

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?

Answer by Ian Agol for Codimension zero immersions

2011-06-17T02:05:52Z

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.

Answer by Ulrich Bauer for Codimension zero immersions

2011-06-28T08:03:30Z

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 $F|_{\partial N}=f$.