Compact Lie group action and equivariant bordism

Abstract:

Let $G$ be a compact Lie group and $H$ a compact Lie subgroup of $G$ contained in the center of $G$ with ${H^m}$ the maximal subgroup in the center, $H$ being $H$-boundary. Let $pr:{H^m} \to H$ be the projection onto the $r$th factor and ${H_r}$ be the $r$th factor of ${H^m}$. Let $\{ {L_r}\}$ be a family of subgroups of $G$ such that ${L_r} \cap {H_r}$ is nontrivial. Consider a $G$-manifold ${M^n}$ with $pr({G_x} \cap {H^m})$ trivial or containing ${L_r}$, for every $x$ in ${M^n}$. The main result of the paper is that if $\forall x \in {M^n}$, ${p_r}({G_x} \cap {H^m})$ is trivial at least for one $r$, then ${M^n}$ is a $G$-boundary.