TITLE: A symmetric embedding of manifolds
QUESTION [10 upvotes]: 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 $i\in \{1,2,\ldots ,n\}$?

REPLY [20 votes]: 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." that in  dimensions $\ge 4$ every oriented cobordism class contains such a manifold.