Generalized Schoenflies - formalizing step in proof?

Question

[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]

I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, which is a specific case of the generalized Schoenflies theorem:

Every smoothly embedded $S^2\subset \mathbb{R}^3$ bounds a smooth 3-ball.

The proof seems to rely on intuition for these low-dimensional arguments, which I find disconcerting, because I have not yet developed that intuition, so I am trying to give actual formal proofs for the statements in Hatcher's proof.

The proof begins with a generic smoothly embedded closed surface $S\subset\mathbb{R}^3$. I have been able to prove that I can isotope $S$ so that projection on the last coordinate $\pi:\mathbb{R}^3\rightarrow\mathbb{R}$ is a Morse function on $S$. Hatcher then argues that if $t$ is a regular value for $\pi$, then $\pi^{-1}(t)\cap S$ is a finite collection of circles.

The proof continues by taking an innermost circle $C\subset \pi^{-1}(t)\cap S$, which by 2-dimensional Schoenflies bounds a disk $D$, and $D\cap S=\partial D=C$. Hatcher then uses surgery to cut away a neighborhood of $C$ in $S$, and cap the cuts with two disks.

This last part is what I want to formalize. It seems we are finding a small-enough tubular neighborhood $C\times(-\epsilon,\epsilon)\subset S$, and then removing that, leaving $S_-=C\times\{-\epsilon\}$ and $S_+=C\times\{\epsilon\}$. Again by 2-dimensional Schoenflies, these bound disks $D_-$ and $D_+$. What I don't get is: why are $S_-$ and $S_+$ still innermost? Or, put differently, why is $D_-\cap S=S-$ (and similarly for $S_+$)?

Intuitively, this seems obvious, and it seems like some sort of "continuity" argument would work, but I cannot figure out how to make this formal. I tried proving that in fact all the disks, "stacked" together for the tubular neighborhood, gave a smooth $D\times [-\epsilon,\epsilon]$, but again I find it hard to make topological arguments when one step of the construction is "apply Schoenflies to get a disk". In particular, I can't prove the projection of this "solid neighborhood" to $D$ is continuous.

Does anyone know how to formalize this? Or, even better, a reference where this type of surgery is discussed? I checked a few places, but only found surgery on a single manifold, not the type discussed here, where we're surgering an embedded submanifold in some ambient manifold.

Answer 1

For an interval $[a,b]\subset{\mathbb R}$ in which the height function $f:S\to {\mathbb R}$ has no critical values one obtains a product structure on $f^{-1}([a,b])$ by following flow lines of the gradient vector field of $f$. This vector field on $f^{-1}([a,b])$ can be extended to a vector field on ${\mathbb R}^2\times [a,b]$ with positive $z$-coordinate everywhere in ${\mathbb R}^2\times [a,b]$. To do this one can first extend the gradient vector field on the surface to a tubular neighborhood of the surface via a projection of this neighborhood onto the surface, then use a smooth partition of unity subordinate to the cover of ${\mathbb R}^2\times [a,b]$ by the tubular neighborhood and the complement of the surface to combine the vector field on the neighborhood with the vertical vector field $(0,0,1)$ on the complement of the surface. In formulas the combined vector field would have the form $v=\phi_1 v_1+\phi_2 v_2$ where $v_1$ is the vector field on the neighborhood and $v_2$ is the vector field on the complement of the surface, with the partition of unity functions $\phi_1$ supported in the neighborhood and $\phi_2$ supported in the complement of the surface. The flow lines of this extended vector field $v$ then give a new product structure on ${\mathbb R}^2\times [a,b]$ extending the product structure on $f^{-1}([a,b])$. In other words one has a level-preserving diffeomorphism of pairs $({\mathbb R}^2\times [a,b],f^{-1}([a,b]))\approx ({\mathbb R}^2\times [a,b], f^{-1}(a)\times [a,b])$.

This is a special case of the isotopy extension theorem which says that an isotopy of a submanifold can always be extended to an ambient isotopy of the whole manifold. The proof is essentially the same.

Answer 2

If $t$ is a regular value, then it is a property of Morse functions that there is some small open neighborhood $U$ of $t$ in $\mathbb{R}$ such that $u$ is also a regular value for all $u\in U$. In particular, we can take $U=(t-\delta,t+\delta)$ for some $\delta>0$. But then $\pi^{-1}(U)\cap S\cong(\pi^{-1}(t)\cap S)\times U$, ie. the surface is a product between any two successive critical levels. Now simply choose $\epsilon<\delta$. Then $S_-=(C\times U)\cap\pi^{-1}(t-\epsilon)$ and $S_+=(C\times U)\cap\pi^{-1}(t+\epsilon)$ will be innermost since $C$ is innermost.

Answer 3

You have a collection of circles which move smoothly (hence homotopically, see below) with the height (due to regularity and implicit function theorem). Let us define what means "innermost": It means that if you take a point $x$ from the inner circle then every outer circle will have winding number 1 or -1 around $x$. Or if you take a point $x$ from an outer circle then the inner circle will have winding number 0 around $x$. Now simply use the fact that the winding number is invariant under homotopies, that is, if $H\colon[a,b]\times[0,1]\to\mathbb R^2$ and $x\colon[0,1]\to\mathbb R^2$ are continuous with $H(t,0)=H(t,1)$ and such that $x(t)\notin H(\{t\}\times[0,1])$ for all $t\in[a,b]$ then the winding number $n(H(t,\cdot),x(t))$ is independent of $t\in[a,b]$.