Collapse homotopy for a fixed normal identification

Shrink around $0_S$ so $\psi ={\mathrm{\Phi }}_1^{-1}{\mathrm{\Phi }}_0$ and its inverse are defined. It fixes $0_S$ and induces the identity on the normal quotient by [F1]. For fiber dilation ${\delta }_t$, define ${\psi }_t={\delta }_{1\mathrm{/}t}\psi {\delta }_t$ for $t>0$, and ${\psi }_0=\mathrm{id}$. In bundle charts write $\psi (x,v)=(b(x,v),w(x,v))$, using a target trivialization near $x$. Then $b(x,0)=x$, $w(x,0)=0$ and ${\mathrm{\partial }}_{v}w(x,0)=I$. Taylor's integral formula writes $w(x,tv)\mathrm{/}t={\int }_0^1{\mathrm{\partial }}_{v}w(x,utv)vdu$, while $b(x,tv)\to x$. These formulas prove smooth extension at $t=0$ with value $(x,v)$. They are coordinate-compatible because the dilation is intrinsic.

Apply the same formulas to ${\psi }^{-1}$. Compactness of $S\times I$ gives a single small disk on which the two families and their composites are defined; shrinking again if necessary, their compositions are the identity, by the identity for $t>0$ and continuity at zero. Thus each ${\psi }_t$ is a diffeomorphism onto its image, and ${\mathrm{\Phi }}_1{\psi }_t$ is a smooth path of compatible tubular charts from ${\mathrm{\Phi }}_1$ to the germ of ${\mathrm{\Phi }}_0$. Its induced normal derivative remains the identity: in the above local expression ${\mathrm{\partial }}_v(w(x,tv)\mathrm{/}t){\mathrm{\mid }}_{v=0}=I$, while the horizontal derivative is irrelevant to the normal quotient. A common small closed tube exists by compactness.

Use that common radius to collapse along ${\mathrm{\Phi }}_1{\psi }_t$. The inverse tube coordinates vary smoothly. The track of the closed tubes is a compact subset of $X\times I$, its boundary maps to the basepoint, and closed pasting as in [F2] proves joint continuity; at a compactification point the compact track gives a uniform constant neighborhood. This supplies the based homotopy of the common-radius endpoint collapses. Interpolate each endpoint radius to its original radius within that endpoint chart: the formula is fiber scaling on the varying tube, agrees with the basepoint at its boundary, and is jointly continuous by the same compact-track argument. Radial cutoffs are covered by [F2].

Finally interpolate metrics by $h_t=(1-t)h_0+t{h}_1$, use their canonical radial maps [F3] to express all targets in the $h_0$ model, and choose a common small tube uniformly in $t$. The formulas and pasting of step 3.1 give the corresponding homotopy. Concatenation proves precisely the stated choice independence. If a chart is instead precomposed with a normal automorphism, step 1.1 limits to that automorphism rather than the identity; for a point and a reflected normal line the resulting sphere maps have opposite degree. Hence preserving the specified normal data is essential to this proof and statement.