A meridian-preserving torus homeomorphism extends over a solid torus

Let $c((e^{i\theta },w),s)=((1-s\mathrm{/}8)e^{i\theta },w)$ for $(e^{i\theta },w)\in \mathrm{\partial }{V}^{\mathrm{\prime }}$ and $s\in [0,2]$; this is a collar because its radial coordinate ranges from $1$ to $3\mathrm{/}4$. Set $\rho (s)=1-s$ for $0\le s\le 1$ and $\rho (s)=0$ for $1\le s\le 2$. The map $J(c(x,s))=c(H_{\rho (s)}(x),s)$ on the collar, extended by the identity outside it, is a homeomorphism: its inverse replaces $H_{\rho (s)}$ with $H_{\rho (s)}^{-1}$. It restricts to $H_1$ at the boundary and is isotopic to the identity through the same formula with $H_{u\rho (s)}$ for $u\in [0,1]$. Thus $h_0=J{\mathrm{\mid }}_{\mathrm{\partial }{V}^{\mathrm{\prime }}}\circ h$ sends $\mu$ exactly to ${\mu }^{\mathrm{\prime }}$, and extending $h_0$ suffices because postcomposition with $J^{-1}$ recovers an extension of $h$ without changing its orientation sign.

Cut $V$ and $V^{\mathrm{\prime }}$ along the product meridian disks $D=D^2\times \{1\}$ and $D^{\mathrm{\prime }}=D^2\times \{1\}$. Each cut-open solid torus is $D^2\times [0,1]$. After centering the interval at $0$, this convex cylinder is homeomorphic to the unit ball by radial rescaling: in direction $u=(u_1,u_2,u_3)\in S^2$ its boundary radius is $R(u)=\min \{(u_1^2+u_2^2{)}^{-1\mathrm{/}2},(2\mathrm{\mid }{u}_3\mathrm{\mid }{)}^{-1}\}$, with a zero denominator interpreted as $+\mathrm{\infty }$, and $(r,u)\mapsto rR(u)u$ gives the homeomorphism. The boundary of each cut-open torus consists of the annulus obtained by cutting the boundary torus along the meridian and two disk faces. The restriction of $h_0$ maps the annular portions to one another. Its common restriction to $\mu$ extends over each disk face by [F1], using the same disk map on both copies. The radial disk maps send the center of each disk face to the center of its target face. These maps assemble to a homeomorphism $\psi :\mathrm{\partial }B\to \mathrm{\partial }{B}^{\mathrm{\prime }}$ between the sphere boundaries of the cut-open balls, carrying the pair of disk-face centers to that pair in the target.

Identify each cut-open ball with the unit ball in ${\mathbb{R}}^3$. Apply [F1] with $d=3$ to extend $\psi$ radially over the ball. The cylinder-to-ball radial rescaling of step 2.1 carries the central product arc $\{0\}\times [0,1]$ to the diameter joining the two disk-face center points. Since $\psi$ carries those endpoints to the target pair, its radial extension carries this entire diameter to the corresponding target diameter. Thus the ball extension preserves the product core arc. The extension has orientation sign equal to the degree sign of $\psi$. On the open annulus away from the disk faces, $\psi$ is exactly $h_0$; cutting does not change the local boundary orientation there, so this degree sign is the orientation sign of $h_0$.

The two copies of each meridian disk receive the same disk map from [F1], so the ball extension agrees on their identified faces. It therefore descends after regluing to a homeomorphism $V\to V^{\mathrm{\prime }}$ extending $h_0$ and carrying the product core circle to the target core circle. Composing with $J^{-1}$ extends the original $h$; this collar-supported map fixes the core and keeps the same orientation sign because $J$ is isotopic to the identity.