Codimension zero immersions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:34:13Z http://mathoverflow.net/feeds/question/67961 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67961/codimension-zero-immersions Codimension zero immersions unknown (google) 2011-06-16T15:43:49Z 2011-06-28T08:03:30Z <p>Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?</p> <p>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?</p> http://mathoverflow.net/questions/67961/codimension-zero-immersions/67979#67979 Answer by Ryan Budney for Codimension zero immersions Ryan Budney 2011-06-16T17:48:21Z 2011-06-17T01:10:07Z <p>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. </p> <p>Given an immersion $$S^{n-1} \to \mathbb R^n$$</p> <p>you get a bundle monomorphism </p> <p>$$TS^{n-1} \to \mathbb R^n$$</p> <p>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 $TS^{n-1} \to \mathbb R^n$. </p> <p>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. </p> <p>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? </p> http://mathoverflow.net/questions/67961/codimension-zero-immersions/68021#68021 Answer by Ian Agol for Codimension zero immersions Ian Agol 2011-06-17T02:05:52Z 2011-06-17T03:29:01Z <p>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. <a href="http://www.ams.org/mathscinet-getitem?mr=2616455" rel="nofollow">Samuel Blank</a> found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposition has been given by <a href="http://archive.numdam.org/article/SB_1966-1968__10__473_0.pdf" rel="nofollow">Valentin Poenaru</a>, and the criterion has been extended to $S^2$ by <a href="http://arxiv.org/abs/1012.4923" rel="nofollow">Frisch</a>. There is also a bit of discussion in these papers about the higher dimensional problem. </p> http://mathoverflow.net/questions/67961/codimension-zero-immersions/69009#69009 Answer by Ulrich Bauer for Codimension zero immersions Ulrich Bauer 2011-06-28T08:03:30Z 2011-06-28T08:03:30Z <p><a href="http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01572-3/" rel="nofollow">Christian Pappas</a> 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 $F|_{\partial N}=f$.</p>