Seam–interior smooth compatibility for the two-disk quotient

Let $m\in\mathbb N$, let $E_m=\mathbb R^{m+1}$, and put $D^{m+1}=\{w\in E_m:\|w\|\leq1\}$ and $S^m=\{u\in E_m:\|u\|=1\}$, with their Euclidean subspace topologies. Give $S^m$ its standard stereographic smooth structure. Fix a homeomorphism $\varphi:S^m\to S^m$. Let

have the quotient topology, where the displayed pairs generate the equivalence relation. Write $[w]_L,[w]_R$ for the quotient images, $V_L=\{[w]_L:\|w\|<1\}$, and $V_R=\{[w]_R:\|w\|<1\}$.

The open seam $U_\varphi$ is the image of the explicit bicollar

This is the source's homeomorphism onto the open seam; $U_\varphi,V_L,V_R$ are open and cover the quotient. The two interiors are disjoint. The quotient seam itself corresponds to $t=0$.

For each $x\in U_\varphi$, write $c_\varphi^{-1}(x)=(u_x,t_x)$. Let $\sigma_{u_x}$ be the standard stereographic chart selected at $u_x$, including the source's orthonormal-basis identification with $\mathbb R^m$, and let $A_m:\mathbb R^m\times\mathbb R\to E_m$ be the continuous linear identification $A_m(a,t)=(t,a)$. The seam regional chart $\chi_x$ is specified by

on its chart source. For every $y_L\in V_L$ and $y_R\in V_R$, the corresponding regional charts $\lambda_{y_L}$ and $\rho_{y_R}$ give the original Euclidean disk-interior coordinates, $\lambda_{y_L}([w]_L)=w$ and $\rho_{y_R}([w]_R)=w$. These regional charts are partial homeomorphisms from the actual quotient to $E_m$, obtained using the open-region inclusions.

For any two such regional charts $a,b$, define $T_{a\to b}=b\circ a^{-1}$ only on its natural overlap domain $a(\operatorname{dom}a\cap\operatorname{dom}b)$. Every smoothness assertion below is $C^\infty$ over $\mathbb R$ on exactly that domain, including the case of an empty overlap. It does not assert smoothness of a totalized inverse outside the domain. All dimensions $m\geq0$ are included.

For every $m\in\mathbb N$ and every homeomorphism $\varphi:S^m\to S^m$, the explicit regional charts on $Q_\varphi$ satisfy all four assertions: (i) for every $x\in U_\varphi,y_L\in V_L$, the seam-to-left-interior transition is $C^\infty$ on its natural overlap source; (ii) for every such $x,y_L$, the reverse transition is $C^\infty$ on its natural overlap source; (iii) if $\varphi^{-1}$ is smooth between the standard spheres, then for every $x\in U_\varphi,y_R\in V_R$, the right-interior-to-seam transition is $C^\infty$ on its natural overlap source; and (iv) if $\varphi$ is smooth between the standard spheres, then for every such $x,y_R$, the seam-to-right-interior transition is $C^\infty$ on its natural overlap source. The smoothness assumptions in (iii) and (iv) remain inside their respective implications. In particular, a smooth boundary diffeomorphism supplies both assumptions, without changing the stronger unconditional left-side assertions.

For $u$ in the sphere chart source, the geometric overlap formulas, before application of $\sigma_{u_x}$ and $A_m$, are

In both inverse formulas, $1/2<\|w\|<1$. The actual inverse overlap additionally requires $w/\|w\|$ on the left, or $\varphi^{-1}(w/\|w\|)$ on the right, to belong to the source of $\sigma_{u_x}$. Thus the domain must not be replaced by the whole annulus. These formulas identify the coordinate changes being asserted smooth; they are not a proof explanation.

This single target groups four proved assertions from the cited source. It concerns actual regional chart transitions; it does not assert a global smooth-manifold instance, a diffeomorphism with the standard sphere, or a decomposition of an arbitrary homotopy sphere into two standard disks.