Topologies and sheaves on causal manifolds

Abstract

A causal manifold $(M,{\hat{I}}^3)$ is a manifold $M$ endowed with a closed proper cone ${\hat{I}}^3$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of ${\hat{I}}^3$. Let $\hat{I}»$ be the antipodal of the polar cone of ${\hat{I}}^3$. An open set $U$ of $M$ is called ${\hat{I}}^3$-open if its Whitney normal cone contains the interior of ${\hat{I}}^3$. Similarly, $U$ is called $\hat{I}»$-open if the micro-support of the constant sheaf on $U$ is contained in $\hat{I}»$. We begin by proving that the two notions coincide. Next, we prove that if $(M,{\hat{I}}^3)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $\hat{I}»$ and the derived category of sheaves on the manifold $M$ endowed with the ${\hat{I}}^3$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.

Corollary 5.8 of the previous version being false, it has been deleted. Other minor corrections