Disk bundle, sphere bundle, and Thom space: the differential topology interface

For a supplied metric $h$ on a finite-rank real vector bundle $E\to B$, write $D_h(E)=\{v\in E:\mathrm{\parallel }v{\mathrm{\parallel }}_h\le 1\},S_h(E)=\{v\in E:\mathrm{\parallel }v{\mathrm{\parallel }}_h=1\},{\mathrm{Th}}_h(E)=D_h(E)\mathrm{/}{S}_h(E)\mathrm{.}$ This is exactly Disk, sphere, and Thom spaces of a metric vector bundle, with the same based quotient convention $X\mathrm{/}\mathrm{\varnothing }=X_{+}$. This item supplies DT notation, not a second definition. Quotients are formed in compactly generated Hausdorff spaces; the compact smooth bases of the geometric applications need no change of topology.

The complement of the Thom basepoint is the image of the open disk bundle $D_h^{\circ }(E)=\{v:\mathrm{\parallel }v{\mathrm{\parallel }}_h<1\}$: in the quotient where $S_h(E)$ is collapsed, the set $D_h^{\circ }(E)=D_h(E)\setminus S_h(E)$ is saturated and the quotient map restricts to a homeomorphism of it onto the complement of the basepoint; in the case $S_h(E)=\mathrm{\varnothing }$ the complement of the added point is $D_h(E)=D_h^{\circ }(E)$. The fiberwise radial expansion $e(v)=\frac{v}{\sqrt{1+\mathrm{\parallel }v{\mathrm{\parallel }}_h^2}}(v\in E),$ with inverse $w\mapsto w\mathrm{/}\sqrt{1-\mathrm{\parallel }w{\mathrm{\parallel }}_h^2}$ on $D_h^{\circ }(E)$, is a homeomorphism $E\to D_h^{\circ }(E)$. When $E$ is smooth, transport its smooth structure along this homeomorphism to the nonbasepoint stratum. If $h$ is also smooth, both formulas are smooth in the original bundle coordinates, so this is the usual open-submanifold smooth structure on $D_h^{\circ }(E)$. A merely continuous metric does not imply that regularity; smooth tubular applications below supply a smooth metric. No smooth manifold structure at the Thom basepoint is presumed, and in rank zero the expansion is the identity $B\to D_h^{\circ }(E)=E$.