The combinatorics and topology of proper toric maps

We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the $f$-vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of $Y$, which is positive precisely when the corresponding closed subset of $Y$ appears as a support in the Decomposition Theorem. The description of this invariant involves the stalks of the intersection cohomology complexes on $X$ and $Y$, but in the case when both $X$ and $Y$ are simplicial, there is a simple formula in terms of the relative $f$-vector.