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 ${\Phi }^{\ast }$ of $G$ on the reduced leafwise cohomology $\bar{H}(\mathcal{F})$. By using leafwise Hodge theory, the supertrace of ${\Phi }^{\ast }$ 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.

arXiv comment40 pages, LaTeX 2e