Assume that $M$ is a manifold.


Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x\_{1},x\_{2},\ldots x\_{i},\ldots,x\_{n}) \mapsto (x\_{1},x\_{2},\ldots , -x\_{i},\ldots,x\_{n})$, for all $i\in \{1,2,\ldots ,n\}$?



A symmetric embedding of manifolds
No. There exist closed manifolds which do not admit any compact group actions, in particular, no $Z\_2$ actions. For example, Shultz showed in ["Group actions on hypertoral manifolds. II."](https://eudml.org/doc/152358) that in dimensions $\ge 4$ every oriented cobordism class contains such a manifold.


