Pontryagin–Thom collapse with specified normal data

Let $S\subset X$ be a compact embedded smooth submanifold without boundary, of codimension $r$, in a smooth boundaryless manifold $X$. A normal datum on $(S,X)$ is a smooth real vector bundle $E\to S$ of rank $r$ together with a smooth bundle isomorphism $\alpha :E\to TX{\mathrm{\mid }}_S\mathrm{/}TS$ onto the normal quotient of Normal and conormal bundles of an embedded submanifold.

A compatible tubular chart for the datum $(E,\alpha )$ is a diffeomorphism $\mathrm{\Phi }$ of a neighbourhood of the zero section $0_S$ in $E$ onto a neighbourhood of $S$ in $X$, with $\mathrm{\Phi }(s,0)=s$ for every $s$, whose induced map on the normal quotient is precisely $\alpha$: identifying the vertical subspace of $T_{(s,0)}E$ with $E_s$, the composite $E_s\stackrel{d{\mathrm{\Phi }}_{(s,0)}{\mathrm{\mid }}_{E_s}}{\to }{T}_{s}X\stackrel{q_s}{\to }{T}_{s}X\mathrm{/}{T}_{s}S=\nu (S{)}_s$ equals ${\alpha }_s$. Fixing the zero section alone is not the compatibility condition; when $E$ is the normal quotient itself compatibility says that this induced map is the identity.

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Compatible charts exist: the submanifold $S$ is closed in the Hausdorff manifold $X$ because it is compact, so Compatible tubular charts realize a prescribed normal identification applies to $i:S\hookrightarrow X$ and the supplied smooth datum $(E,\alpha )$ and produces such a chart. This inherited hypothesis is the only choice used here: once a chart, a metric and a radius have been supplied, the collapse formula below selects nothing.

Supply a smooth metric $h$ on $E$ and a radius $\rho >0$ such that $\mathrm{\Phi }$ is defined on a neighbourhood of $D_{\rho }(E)$. The collapse $c_{\mathrm{\Phi },\rho }:X_{+}\to {\mathrm{Th}}_h(E)$ of Disk bundle, sphere bundle, and Thom space: the differential topology interface sends $\mathrm{\Phi }(s,v)$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_h<\rho$ to the class of $(s,v\mathrm{/}\rho )$, and sends every other point of $X_{+}$ to the Thom basepoint. This is well defined because $\mathrm{\Phi }$ is injective, and it is based because the disjoint basepoint is not in the tube. If $X$ is locally compact, the same formula defines $c:X^{+}\to {\mathrm{Th}}_h(E)$ on the one-point compactification, since $\mathrm{\Phi }(D_{\rho }(E))$ is compact and $c$ is constant off that compact set. The metric, chart and radius are auxiliary choices within the specified normal-data class; the normal identification is part of the input.