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Let $G$ be a compact Lie group, and $M$ be a smooth $n$-dimensional $G$-manifold which admits an orientation preserving the $G$-action. Then $M$ has a natural Whitney stratification induced by the orbit types decomposition $$M=\bigsqcup_{H} M_{(H)} .$$ In particular, there is no stratum with codimension one. If $X\subset M$ is a $G$-invariant sub-Whitney stratification with dimension $k$, i.e. every stratum of $X$ is a $G$-invariant submanifold of $M$. Suppose that $X$ is $G$-oriented, i.e. the top stratum admits an orientation preserving by the group action, then do there exist stratum of codimension one in $X$?

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