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An ambient isotopy preserves the orientation of an invariant round sphere

Statement

Let $H : R^{3} \times I \to R^{3}$ be an ambient isotopy with $H_{0} = i d$ (Smooth isotopies, diffeotopies and ambient isotopies) and suppose $H_{1}$ maps the closed unit ball $B^{3} \subseteq R^{3}$ to itself. Then $H_{1}$ restricts to a diffeomorphism of the unit sphere $S^{2} = \partial B^{3}$ of degree $+ 1$ ; that is, $H_{1} ∣_{S^{2}}$ preserves the boundary orientation of $S^{2}$ (Induced boundary orientation). Consequently, if $r : R^{3} \to R^{3}$ is a linear reflection in a plane through the origin (so $\det r = - 1$ and $r$ preserves $S^{2}$ ), there is no such ambient isotopy with $H_{1} ∣_{S^{2}} = r ∣_{S^{2}}$ .

Facts & Assumptions

Given: An ambient isotopy $H : R^{3} \times I \to R^{3}$ with $H_{0} = i d$ and $H_{1} (B^{3}) = B^{3}$ .

[F1]

Each $H_{t}$ is a diffeomorphism of $R^{3}$ and $H$ is smooth; in particular every differential $d H_{t} (x)$ is an invertible linear map (Smooth isotopies, diffeotopies and ambient isotopies, Diffeomorphisms and local diffeomorphisms of manifolds).

[F2]

The entries of the Jacobian matrix of the smooth map $(x, t) \mapsto H_{t} (x)$ are partial derivatives of a $C^{\infty}$ function of several variables and are therefore continuous; the determinant is a polynomial in the matrix entries ( $C^{k}$ maps and multi-index derivative notation in Euclidean space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries).

[L1]

A continuous real function on $I = [0, 1]$ that never vanishes and is positive at $0$ is positive everywhere (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ).

[L2]

A diffeomorphism of manifolds with boundary maps the boundary onto the boundary and the interior onto the interior (Diffeomorphisms preserve interior and boundary).

[L3]

The boundary orientation of $S^{2} = \partial B^{3}$ is the outward-normal-first orientation of Induced boundary orientation applied to the oriented closed ball; $S^{2}$ is a nonempty connected orientable boundaryless manifold (Orientable manifolds).

[L4]

A diffeomorphism between nonempty connected oriented boundaryless manifolds has degree $+ 1$ if it preserves orientation and $- 1$ if it reverses it (Degree of an orientation-preserving or reversing diffeomorphism). A linear reflection $r$ with $\det r = - 1$ restricts to a diffeomorphism of $S^{2}$ that reverses the outward-normal-first boundary orientation, since $r$ maps the ball onto itself, carries outward normals to outward normals, and reverses the ambient orientation; hence $r ∣_{S^{2}}$ has degree $- 1$ by the same proposition.

Proof

technique · direct

1.1F1F2L1

Fix $x \in R^{3}$ and put $φ (t) : = \det d H_{t} (x)$ for $t \in I$ . The function $φ$ is continuous on $I$ by [F2], and it never vanishes because each $d H_{t} (x)$ is invertible by [F1]; since $φ (0) = \det i d = 1 > 0$ , the intermediate value property [L1] gives $φ (t) > 0$ for every $t \in I$ . Hence $d H_{t} (x)$ is orientation-preserving at every $(x, t)$ , and in particular $H_{1}$ preserves the orientation of $R^{3}$ at every point.

2.1F1L2step 1.1

Since $H_{1}$ is a diffeomorphism and $H_{1} (B^{3}) = B^{3}$ , [L2] shows that $H_{1}$ maps the interior of $B^{3}$ onto the interior and the sphere $S^{2} = \partial B^{3}$ onto itself; thus $H_{1} ∣_{S^{2}} : S^{2} \to S^{2}$ is a diffeomorphism. Moreover $d H_{1}$ carries the outward transverse direction of $S^{2}$ at each $p$ to the outward transverse direction at $H_{1} (p)$ : the interior maps to the interior, so a tangent vector pointing into the ball maps to a vector pointing into the ball, and an outward transverse vector maps to an outward transverse vector: the inward boundary coordinate of the image vanishes at the boundary, is positive on the interior side, and has nonzero normal derivative by invertibility, so that derivative is positive. Orthogonality to the sphere need not be preserved.

3.1L3L4step 1.1step 2.1

The restriction $H_{1} ∣_{S^{2}}$ preserves the outward-normal-first boundary orientation of [L3]: if $(v_{1}, v_{2})$ is a positive basis of $T_{p} S^{2}$ , so that $(n_{p}, v_{1}, v_{2})$ is a positive basis of $T_{p} R^{3}$ with $n_{p}$ outward, then step 1.1 makes $(d H_{1} (n_{p}), d H_{1} (v_{1}), d H_{1} (v_{2}))$ positive at $H_{1} (p)$ , and $d H_{1} (n_{p})$ is an outward vector by step 2.1, so $(d H_{1} (v_{1}), d H_{1} (v_{2}))$ is a positive basis of $T_{H_{1} (p)} S^{2}$ . Hence $H_{1} ∣_{S^{2}}$ is an orientation-preserving diffeomorphism of $S^{2}$ , and [L4] gives $\deg (H_{1} ∣_{S^{2}}) = + 1$ .

4.1L4step 3.1∎

Let $r : R^{3} \to R^{3}$ be a linear reflection, $\det r = - 1$ , preserving $S^{2}$ . By [L4] the restriction $r ∣_{S^{2}}$ reverses the boundary orientation and has degree $- 1$ , while every diffeomorphism $H_{1} ∣_{S^{2}}$ arising as above has degree $+ 1$ by step 3.1; therefore $H_{1} ∣_{S^{2}} = r ∣_{S^{2}}$ is impossible, and no such ambient isotopy can restrict to the reflection.

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