Top normal classes vanish for Euclidean embeddings

Given: A smooth embedding $j:M^m\hookrightarrow {\mathbb{R}}^{m+k}$ of a closed smooth manifold with $m\ge 1$, $k\ge 1$, its normal bundle ${\nu }_j$ of rank $k$, and AC (The Axiom of Choice).

Under ${\mathrm{A}\mathrm{C}}_{\omega }$ (hence under AC) the embedding gives a rank-$k$ stable normal inverse $({\nu }_j,\phi )$ of $M$, where ${\nu }_j=j^{\ast }T{\mathbb{R}}^{m+k}\mathrm{/}dj(TM)$ is the normal quotient (An embedding into Euclidean space gives a rank-(n-m) stable normal inverse); if ${\nu }_j$ is integrally oriented it is an oriented rank-$k$ bundle in the sense of the Thom interface.

Let $\mathrm{\Phi }$ be a compatible tubular chart for ${\nu }_j$ with a metric $h$ and radius $\rho$ (existing under countable choice, hence under AC); its collapse is a based continuous map $c:({\mathbb{R}}^{m+k}{)}^{+}\to {\mathrm{Th}}_h({\nu }_j)$ sending the tube to the disk-sphere quotient model and every other point to the basepoint. The zero section $s:M\to D({\nu }_j)$ followed by the quotient map $q:D({\nu }_j)\to {\mathrm{Th}}_h({\nu }_j)$ is the based zero section $z=q\circ s$, and on $M$ the collapse satisfies $c\circ j=z$ (Pontryagin–Thom collapse with specified normal data).

For either coefficient ring $R={\mathbb{F}}_2$ or $R=\mathbb{Z}$ with a supplied integral orientation, the normalized Thom class $u\in H^k(D({\nu }_j),S({\nu }_j);R)$ corresponds under the quotient identification to a class $u\in H^k({\mathrm{Th}}_h({\nu }_j);R)$ of positive degree, and the quotient-map pullback $q^{\ast }(u)$ is the relative-to-absolute image of the relative Thom class; hence $z^{\ast }(u)=s^{\ast }{q}^{\ast }(u)=e({\nu }_j)$, the Euler class of Euler class by zero-section pullback of the Thom class, which by The mod-two Euler class is the top Stiefel–Whitney class equals $w_k({\nu }_j)$ for $R={\mathbb{F}}_2$, while for $R=\mathbb{Z}$ it is the oriented Euler class (Thom class and Thom isomorphism: the AT interface, Euler class by zero-section pullback of the Thom class, The mod-two Euler class is the top Stiefel–Whitney class).

Since $m\ge 1$ and $k\ge 1$, one has $0<k<N$ for $N=m+k$, so the one-point compactification has $H^k(({\mathbb{R}}^N{)}^{+};R)=0$ for $R=\mathbb{Z}$ and $R={\mathbb{F}}_2$ (Positive intermediate cohomology of compactified Euclidean space vanishes). Cohomology is contravariantly functorial, so $f^{\ast }(0)=0$ and $(c\circ j{)}^{\ast }=j^{\ast }{c}^{\ast }$ (Singular cohomology is contravariantly functorial).

A closed smooth manifold is a paracompact Hausdorff CGWH space of CW homotopy type over which every smooth bundle, in particular ${\nu }_j$, is numerable (Smooth manifolds have CW homotopy type); this places $M$ and ${\nu }_j$ in the scope of the Thom interface and of the mod-two Euler class theorem.

For the stable normal inverse $({\nu }_j,\phi )$ one has $w({\nu }_j)=w(TM{)}^{-1}$, so $w_k({\nu }_j)={\bar{w}}_k(TM)$; also $w_i({\nu }_j)=0$ for $i>k$ because ${\nu }_j$ has rank $k$ (The normal Stiefel-Whitney class is the multiplicative inverse of the tangent class, Stiefel–Whitney classes from the projective-bundle relation).