Convex polytopes and linear algebra

Abstract

This paper defines, for each convex polytope $Δ$, a family $H_wΔ$ of vector spaces. The definition uses a combination of linear algebra and combinatorics. When what is called exact calculation holds, the dimension $h_wΔ$ of $H_wΔ$ is a linear function of the flag vector $fΔ$. It is expected that the $H_wΔ$ are examples, for toric varieties, of the new topological invariants introduced by the author in "Local-global intersection homolog" (preprint alg-geom/9709011).