2 clarified non-orientable n/2 case

For which values of $n$ does there exist an embedding of a smooth compact manifold $M\hookrightarrow R^n$ into $n$-dimensional Euclidean space such that the normal bundle to $M$ has no nonvanishing section?

If such an embedding does exist, can $M$ be taken to be orientable? What can be the dimension of $M$?

In other words, I'm asking for submanifolds $M$ of $R^n$ which cannot be "immediately pushed off of themselves" by the flow of a vector field. Conversely, a theorem of Hirsch states that if a nonvanishing normal field does exist then $M$ immerses in $R^{n-1}$.

If $n=4k$ for some integer $k$, then one can find non-orientable embedded $2k$-submanifolds $M$ in $R^{4k}$ with which have no such sections---the easiest (to me) way that I know of doing this is to ask for $M$ to be Lagrangian with respect to the standard symplectic structure, so that its normal bundle will be isomorphic to its tangent bundle and so one just needs $M$ to have nonzero Euler characteristic (since then the (integral) twisted Euler class in the local coefficient cohomology associated to $w_1$ will be nontrivial), and then examples can be constructed by Lagrangian surgery. On the other hand an orientable submanifold of $R^n$ of dimension $\frac{n}{2}$ will always have a nonvanishing normal vector field, since the only obstruction to constructing such is the Euler class of the normal bundle, which is the restriction of a cohomology class from $R^n$ and so vanishes.

If $\dim M>\frac{n}{2}$ then there are higher-order obstructions to the existence of a nonvanishing section of the normal bundle, which can be nontrivial in the orientable case. There are some examples of this described by Massey; for instance for $s\geq 1$, $CP^{2^{s}}$ embeds in $R^{4\cdot2^s-1}$, and there is no normal section because the secondary obstruction is nontrivial. I think (though haven't checked carefully) that a product of two of these $CP^{2^s}$ embeddings should also lack a nonvanishing normal vector field, due to product formulas for the obstruction classes. But these examples seem rather special, and I haven't been able to see a way of adapting the ideas underlying them to construct examples in arbitrary ambient dimension.

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# Embeddings without nonvanishing normal vector fields

For which values of $n$ does there exist an embedding of a smooth compact manifold $M\hookrightarrow R^n$ into $n$-dimensional Euclidean space such that the normal bundle to $M$ has no nonvanishing section?

If such an embedding does exist, can $M$ be taken to be orientable? What can be the dimension of $M$?

In other words, I'm asking for submanifolds $M$ of $R^n$ which cannot be "immediately pushed off of themselves" by the flow of a vector field. Conversely, a theorem of Hirsch states that if a nonvanishing normal field does exist then $M$ immerses in $R^{n-1}$.

If $n=4k$ for some integer $k$, then one can find non-orientable embedded $2k$-submanifolds $M$ in $R^{4k}$ with which have no such sections---the easiest (to me) way that I know of doing this is to ask for $M$ to be Lagrangian with respect to the standard symplectic structure, so that its normal bundle will be isomorphic to its tangent bundle and so one just needs $M$ to have nonzero Euler characteristic, and then examples can be constructed by Lagrangian surgery. On the other hand an orientable submanifold of $R^n$ of dimension $\frac{n}{2}$ will always have a nonvanishing normal vector field, since the only obstruction to constructing such is the Euler class of the normal bundle, which is the restriction of a cohomology class from $R^n$ and so vanishes.

If $\dim M>\frac{n}{2}$ then there are higher-order obstructions to the existence of a nonvanishing section of the normal bundle, which can be nontrivial in the orientable case. There are some examples of this described by Massey; for instance for $s\geq 1$, $CP^{2^{s}}$ embeds in $R^{4\cdot2^s-1}$, and there is no normal section because the secondary obstruction is nontrivial. I think (though haven't checked carefully) that a product of two of these $CP^{2^s}$ embeddings should also lack a nonvanishing normal vector field, due to product formulas for the obstruction classes. But these examples seem rather special, and I haven't been able to see a way of adapting the ideas underlying them to construct examples in arbitrary ambient dimension.