Le $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ the $G$-fixed subset. In general, $F$ has finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation-preserving $G$-action, and the fixed submanifold has positive dimension, then why the codimension of the fixed submanifold is even?

Let $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ the $G$-fixed subset. In general, $F$ has finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation-preserving $G$-action, and the fixed submanifold has positive dimension, then why the codimension of the fixed submanifold is even?

Le $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ be the $G$-fixed subset. In general, $F$ have finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation preserved by the $G$-action, and the fixed submanifold have positive dimension, then why the codimension of fixed submanifold is even?

Le $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ the $G$-fixed subset. In general, $F$ has finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation-preserving $G$-action, and the fixed submanifold has positive dimension, then why the codimension of the fixed submanifold is even?

Le $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ be the $G$-fixed subset. In general, $F$ have finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation preserved by the $G$-action, then why the codimension of fixed submanifold is even?

Le $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ be the $G$-fixed subset. In general, $F$ have finitely many connected components and each component is a submanifold of $M$. My question is: if $M$ admits an orientation preserved by the $G$-action, and the fixed submanifold have positive dimension, then why the codimension of fixed submanifold is even?