Topological model for h"-vectors of simplicial manifolds

TL;DR

This paper introduces a topological model linking the h''-vector of simplicial manifolds to the Poincare duality algebra, providing new insights into their nonnegativity and symmetry properties.

Contribution

It constructs a manifold with boundary from a simplicial poset, establishing a topological framework for understanding h''-vectors and their properties.

Findings

Proves nonnegativity of h''-numbers for simplicial manifolds.

Establishes symmetry h''_k = h''_{n-k} using topological methods.

Provides a topological interpretation of known combinatorial facts.

Abstract

Any manifold with boundary gives rise to a Poincare duality algebra in a natural way. Given a simplicial poset $S$ whose geometric realization is a closed orientable homology manifold, and a characteristic function, we construct a manifold with boundary such that graded components of its Poincare duality algebra have dimensions $h_k"(S)$. This gives a clear topological evidence for two well-known facts about simplicial manifolds: the nonnegativity of $h"$-numbers (Novik--Swartz theorem) and the symmetry $h"_k=h"_{n-k}$ (generalized Dehn--Sommerville relations).