Return to Answer

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems* to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

$^*$ You have maps of vector bundles $\theta_\alpha:{\underline{\mathbb{R}}}_{U_\alpha}^r\to E|_{U_\alpha}$ which you glue by a partition of unity $\{p_\alpha\}$, to obtain that $E$ is generated by global sections, by the map $\theta=\sum_\alpha p_\alpha \cdot\theta_\alpha:\prod^{t(E)}\underline{\mathbb{R}}^r\to E\to 0$. Since $E$ is isomorphic to its dual, you can dualize the map and obtain an inclusion of $E$ into a trivial v.b.

Remark: I haven't thought if this would give, in the case of a total space of a v.b., a better result than Cohen's embedding theorem itself applied to the projectivization $\mathbb{P}(E\oplus\mathbb{R})$ (or the one-point compactification of $E$ along the fibers) which has dimension $N=n+r$. One should ask if $2n-a(n)+r(n+1)< 2N-a(N)$ for some $n,r$.

I don't know if the following remarks can be considered "of this kind".

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems* to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

$^*$ You have maps of vector bundles $\theta_\alpha:{\underline{\mathbb{R}}}_{U_\alpha}^r\to E|_{U_\alpha}$ which you glue by a partition of unity $\{p_\alpha\}$, to obtain that $E$ is generated by global sections, by the map $\theta=\sum_\alpha p_\alpha \cdot\theta_\alpha:\prod^{t(E)}\underline{\mathbb{R}}^r\to E\to 0$. Since $E$ is isomorphic to its dual, you can dualize the map and obtain an inclusion of $E$ into a trivial v.b.

Remark: I haven't thought if this would give, in the case of a total space of a v.b., a better result than Cohen's embedding theorem itself applied to the projectivization $\mathbb{P}(E\oplus\mathbb{R})$ (or the one-point compactification of $E$ along the fibers) which has dimension $N=n+r$. One should ask if $2n-a(n)+r(n+1)< 2N-a(N)$ for some $n,r$.

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems* to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

$^*$ You have maps of vector bundles $\theta_\alpha:{\underline{\mathbb{R}}}_{U_\alpha}^r\to E|_{U_\alpha}$ which you glue by a partition of unity $\{p_\alpha\}$, to obtain that $E$ is generated by global sections, by the map $\theta=\sum_\alpha p_\alpha \cdot\theta_\alpha:\prod^{t(E)}\underline{\mathbb{R}}^r\to E\to 0$. Since $E$ is isomorphic to its dual, you can dualize the map and obtain an inclusion of $E$ into a trivial v.b.

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems* to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

$^*$ You have maps of vector bundles $\theta_\alpha:{\underline{\mathbb{R}}}_{U_\alpha}^r\to E|_{U_\alpha}$ which you glue by a partition of unity $\{p_\alpha\}$, to obtain that $E$ is generated by global sections, by the map $\theta=\sum_\alpha p_\alpha \cdot\theta_\alpha:\prod^{t(E)}\underline{\mathbb{R}}^r\to E\to 0$. Since $E$ is isomorphic to its dual, you can dualize the map and obtain an inclusion of $E$ into a trivial v.b.

Remark: I haven't thought if this would give, in the case of a total space of a v.b., a better result than Cohen's embedding theorem itself applied to the projectivization $\mathbb{P}(E\oplus\mathbb{R})$ (or the one-point compactification of $E$ along the fibers) which has dimension $N=n+r$. One should ask if $2n-a(n)+r(n+1)< 2N-a(N)$ for some $n,r$.

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

Let $E$ be the total space of a rank $r$ vector bundle over the compact $n$-manifold $M$. Then, as you said, by Cohen $M\subseteq \mathbb{R}^{2n-a(n)}$.

Also (see Atiyah, K-theory, Corollary 1.4.14) you can realize $E$ as a sub-vector-bundle of a trivial bundle $E\subseteq \underline{\mathbb{R}}^m$ for a suitable $m$. In fact, looking at the proof of the preceding Lemma 1.4.12 it seems* to me that you can take $m$ to be $\leq r\times t(E)$ where $t(E)$ is the minimum number of elements of an open cover of $M$ which is trivializing for $E$. And from this, it seems that $t(E)\leq n+1$.

So, you could embed $E$ in $\mathbb{R}^{2n-a(n)+r\cdot (n+1)}$.

$^*$ You have maps of vector bundles $\theta_\alpha:{\underline{\mathbb{R}}}_{U_\alpha}^r\to E|_{U_\alpha}$ which you glue by a partition of unity $\{p_\alpha\}$, to obtain that $E$ is generated by global sections, by the map $\theta=\sum_\alpha p_\alpha \cdot\theta_\alpha:\prod^{t(E)}\underline{\mathbb{R}}^r\to E\to 0$. Since $E$ is isomorphic to its dual, you can dualize the map and obtain an inclusion of $E$ into a trivial v.b.