The Pontryagin-Thom map of a framed submanifold

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $X$ be closed and smooth and let $(N,\phi )$ be a closed framed codimension-$k$ submanifold, $k\ge 0$ (Framings of a normal bundle). A compatible chart $\mathrm{\Phi }$ for $(\nu (N),\mathrm{i}\mathrm{d})$, a supplied smooth metric and a sufficiently small radius give a collapse $c:X_{+}\to \mathrm{Th}(\nu (N))$ as in Pontryagin–Thom collapse with specified normal data. Compose it with the framing homeomorphism and projection of A framing identifies the Thom target with a sphere smash product: $X_{+}\stackrel{c}{\to }\mathrm{Th}(\nu (N))\stackrel{{\mathrm{\Phi }}_{\phi }}{\to }{N}_{+}\wedge S^k\stackrel{p}{\to }{S}^k\mathrm{.}$ Its based homotopy class is the Pontryagin–Thom class. The map is based at the disjoint point of $X_{+}$; its restriction to $X$ need not preserve any preselected point of $X$.

For $k\ge 1$ we use the following normalized smooth representative $f_{(N,\phi )}$. Fix an orientation-preserving stereographic coordinate $z:S^k\setminus \{\mathrm{\infty }\}\to {\mathbb{R}}^k$ with centre $y_0=z^{-1}(0)$ and positive basis $b_0=(d{z}_{y_0}{)}^{-1}(e_1,\dots ,e_k)$. Write $u={\phi }_s(v)$ in a compatible tube $\mathrm{\Phi }(s,v)$, and choose $r>0$ so that the chart is defined on $\{\mathrm{\mid }u\mathrm{\mid }\le 2r\}$; compactness of $N$ gives such a radius. Using The standard smooth step function, put $a(q)=1-\sigma \text{\hspace{0.17em}⁣}\left(\frac{q-r^2\mathrm{/}4}{3r^2\mathrm{/}4}\right)\mathrm{.}$ Thus $a=1$ for $q\le r^2\mathrm{/}4$, $a>0$ for $q<r^2$ and $a=0$ for $q\ge r^2$. Define $f_{(N,\phi )}(\mathrm{\Phi }(s,v))=z^{-1}\text{\hspace{0.17em}⁣}\left(\frac{u}{a(\mathrm{\mid }u{\mathrm{\mid }}^2)}\right)(\mathrm{\mid }u\mathrm{\mid }<r),$ and send all other points to $\mathrm{\infty }$. Near $N$ the target coordinate is exactly $u$. Near $\mathrm{\mid }u\mathrm{\mid }=r$, the coordinate at infinity obtained by inversion is $a(\mathrm{\mid }u{\mathrm{\mid }}^2)u\mathrm{/}\mathrm{\mid }u{\mathrm{\mid }}^2$, which is smooth and extends by zero because $a$ is smooth and vanishes identically for $\mathrm{\mid }u\mathrm{\mid }\ge r$. Hence the map is smooth everywhere and constant near the boundary of the larger tube. Its centre preimage is exactly $N$.

This representative has the preceding collapse class. Use the framing metric $\mathrm{\mid }v{\mathrm{\mid }}_{\phi }=\mathrm{\mid }{\phi }_s(v)\mathrm{\mid }$. In the disk model of Disk bundle, sphere bundle, and Thom space: the differential topology interface, the smooth profile above has normalized radius $b(t)=\frac{t}{\sqrt{a(t^2{)}^2+t^2}}(0\le t\le r),$ with $b(0)=0$, $b(r)=1$ and $b(t)>0$ for $t>0$. Interpolating $b(t)$ with $t\mathrm{/}r$ gives continuous disk-valued radial maps which agree at the boundary and never acquire an extra centre preimage. Closed pasting gives a based homotopy to the ordinary collapse, as in Continuity and smooth local representatives of collapse. Different metrics are compared by the radial Thom homeomorphisms; the next lemma proves tube independence.

For $k=0$, $N$ is clopen in $X$, and $f_{(N,\phi )}:X\to S^0$ sends $N$ to the nonbasepoint and $X\setminus N$ to the basepoint. For $N=\mathrm{\varnothing }$ it is constant at the basepoint. The framing is fixed data; a reflected framing changes the map by the corresponding sphere reflection. Countable choice is inherited only from the compatible-chart and normal-bundle machinery.