Continuity and smooth local representatives of collapse

On the closed tube the continuous map $\mathrm{\Phi }(s,v)\mapsto [(s,v\mathrm{/}\rho )]$ sends its boundary to the basepoint. On the closed complement of its interior the map is constant. These two closed sets cover $X$ and the definitions agree on their intersection, so the closed pasting lemma [F2] proves continuity on $X$; at a disjoint added basepoint continuity is immediate. At the compactification point, the complement of the compact closed tube is a neighborhood mapped constantly to the basepoint; this proves continuity there.

On the inverse image of the nonbasepoint stratum the formula is a smooth tubular inverse followed by fiber scaling. In a bundle trivialization about zero the map reads $(s,v)\mapsto (s,v\mathrm{/}\rho )$, whose derivative in the fibre directions is ${\rho }^{-1}$ times the identity, so it is a submersion there; identifying the normal quotient of $X$ along $S$ with $E$ by ${\alpha }^{-1}$, that vertical derivative is ${\rho }^{-1}{\alpha }^{-1}$ and its zero fibre is exactly $S$. No smoothness assertion at the Thom basepoint is needed.

More generally let $a:[0,\rho ]\to [0,1]$ be smooth on $[0,\rho )$, positive off zero, equal to $\lambda t$ near zero for $\lambda >0$, and equal to $1$ on a neighborhood of $\rho$. Map $v\mathrm{\ne }0$ to $a(\mathrm{\parallel }v\mathrm{\parallel })v\mathrm{/}\mathrm{\parallel }v\mathrm{\parallel }$ and zero to zero, and then take the quotient; outside the tube use the basepoint. This is continuous by step 1.1 and smooth on its nonbasepoint stratum, including zero because its formula there is $\lambda v$. Convex interpolation between this radius profile and $t\mathrm{/}\rho$ stays positive for $t>0$, is linear with positive coefficient near zero, and equals $1$ at the boundary. The same pasting argument on $X\times I$ proves the based homotopy. Thus a cutoff supplies the contracted smooth representative near regular values without assigning a smooth structure at the collapsed point.