Dehn–Sommerville relations for complexity one torus actions

Dehn–Sommerville relations for simple polytopes are a combinatorial consequence of Poincare duality for toric varieties. Acyclic sponge is a notion used to encode the combinatorial structure of a torus action of complexity one. One can define $h$-numbers of any acyclic sponge, and if the sponge originates from a torus action on a manifold $X$, then $h_i$ equals $\beta_{2i}(X)$, so Poincare duality again implies the analogue of Dehn–Sommerville relations. It was previously conjectured that these relations hold for all acyclic sponges, not only those originating from torus actions on manifolds. In this note, we prove this fact using the machinery of two-parametric generating functions introduced by V. Buchstaber.