Foliation preserving Lie group actions and characteristic classes

Abstract:

Let $\tilde {\mathcal {F}}$ be a codimension $k$ foliation of a manifold $M$ and $\mathcal {F}$ a subfoliation of $\tilde {\mathcal {F}}$ with codimension $q$. Let a Lie group $G$ of dimension $k$ act on $M$ transversally locally freely to $\tilde {\mathcal {F}}$ and preserving $\mathcal {F}$. Let $\mathcal {F}’$ be the foliation determined by $\mathcal {F}$ and the $G$-action. Then we have the following relations between exotic classes of $\mathcal {F}$ and $\mathcal {F}’:{\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = {\alpha _{\mathcal {F}’}}([{\hat c_I}{c_J}])$ for $I = ({i_1}, \ldots ,{i_\lambda })$, $J = ({j_1}, \ldots ,{j_\mu })$, $1 \leqslant {j_\gamma },{j_l} \leqslant q - k$, and ${\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = 0$ otherwise.