Lefschetz distribution of Lie foliations

Abstract

Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural Lie group $G$. Its transverse Lie structure can be considered as a transverse action $\Phi$ of $G$ on $(M,\mathcal F)$; i.e., an ``action'' which is defined up to leafwise homotopies. This $\Phi$ induces an action $\Phi^*$ of $G$ on the reduced leafwise cohomology $\bar H(\mathcal F)$. By using leafwise Hodge theory, the supertrace of $\Phi^*$ can be defined as a distribution $L_{dis}(\mathcal F)$ on $G$ called the Lefschetz distribution of $\mathcal F$. A distributional version of the Gauss-Bonett theorem is proved, which describes $L_{dis}(\mathcal F)$ around the identity element. On any small enough open subset of $G$, $L_{dis}(\mathcal F)$ is described by a distributional version of the Lefschetz trace formula.