Dehn-Sommerville from Gauss-Bonnet (1905.04831v1)

Abstract: We give a zero curvature proof of Dehn-Sommerville for finite simple graphs. It uses a parametrized Gauss-Bonnet formula telling that the curvature of the valuation G to f_G(t)=1+f0 t + ... + fd t^d+1 defined by the f-vector of G is the anti-derivative F of f evaluated on the unit sphere S(x). Gauss Bonnet is then parametrized, f_G(t) = sum_x F_(S(x))(t), and holds for all simplicial complexes G. The Gauss-Bonnet formula chi(G)=sum_x K(x) for Euler characteristic chi(G) is the special case t=-1. Dehn-Sommerville is equivalent to the reflection symmetry f_G(t)+(-1)^d f_G(-1-t)=0 which is equivalent to the same symmetry for F. Gauss-Bonnet therefore relates Dehn-Sommerville for G with Dehn-Sommerville for the unit spheres S(x), where it is a zero curvature condition. A class X_d of complexes for which Dehn-Sommerville holds is defined inductively by requiring chi(G)=1+(-1)^d and S(x) in X_(d-1) for all x. It starts with X_(-1)={{}}. Examples are simplicial spheres, including homology spheres, any odd-dimensional discrete manifold, any even-dimensional discrete manifold with chi(G)=2. It also contains non-orientable ones for which Poincar'e-duality fails or stranger spaces like spaces where the unit spheres allow for two disjoint copies of manifolds with chi(G)=1. Dehn-Sommerville is present in the Barycentric limit. It is a symmetry for the Perron-Frobenius eigenvector of the Barycentric refinement operator A. The even eigenfunctions of A^T, the Barycentric Dehn-Sommerville functionals, vanish on X like 22 f1 - 33 f2 + 40 f3 - 45f4=0 for all 4-manifolds.

• The paper proves the equivalence of Dehn-Sommerville relations to a reflection symmetry in f-vector polynomials using a zero curvature approach.
• It applies a parametrized Gauss-Bonnet theorem to relate vertex curvatures with the Euler characteristic in finite simple graphs.
• The study establishes a recursive framework for complexes that naturally satisfy these relations, enabling new insights in combinatorial topology.

Dehn-Sommerville from Gauss-Bonnet

The paper presents a "zero curvature" proof of the Dehn-Sommerville relations within the context of finite simple graphs, framed through a novel interpretation of the Gauss-Bonnet theorem. The approach utilizes a parametrized version of the Gauss-Bonnet formula, which considers the curvature of a valuation mapped from a graph $G$ to its $f$-vector polynomial $f_G(t)$. This is further linked to its anti-derivative $F$, computed over the unit spheres $S(x)$ of the vertices of $G$. Specifically, the parametrized Gauss-Bonnet theorem can be expressed as $f_G(t) = \sum_x F_{S(x)}(t)$, applicable across Whitney simplicial complexes.

The demonstration hinges on the equivalence between the Dehn-Sommerville relations and a specific reflection symmetry of $f_G(t)$: $f_G(t) + (-1)^d f_G(-1-t) = 0$. Here, $d$ represents the maximal dimension of the simplicial complex. This symmetry in $f_G(t)$ is paralleled by the same symmetry in $F$. The research further explores the recursive definition of a class $\mathcal{X}_d$ of complexes where these relations hold, starting from $\mathcal{X}_{(-1)} = \{ \{\} \}$ and advancing through higher dimensions.

Key Theoretical Contributions

1. Parametrized Gauss-Bonnet Theorem: This newly formulated version maintains the relation of the Euler characteristic $\chi(G)$ with vertex-based curvatures rooted in $f$-vector polynomials, parametrically extended over real variable $t$ with the special case $t = -1$ regressing to the classical discrete setting.
2. Dehn-Sommerville Symmetry: Through algebraic re-interpretation, the paper identifies the Dehn-Sommerville relations as intrinsic symmetries in the polynomial space characterized by the $f$-vector. This affirms the symmetry as equivalent to the palindromic nature of the $h$-vector used extensively in manifold theory.
3. Recursive Definition of Complexes: The establishment of $\mathcal{X}_d$ induces a conceptual framework where spheres and specific manifold types naturally satisfy Dehn-Sommerville under a zero curvature insight, contingent upon recursive structural properties.

Implications and Future Directions

The implications are multi-fold, affecting both the theoretical constructs and computational aspects related to graph theory, combinatorics, and topology:

• PALINdromic Structures: The reinforcement of the Dehn-Sommerville symmetry through polynomial analysis provides mathematicians with tools to identify and classify polyhedral structures and other geometric objects with enhanced symmetry considerations.
• Curvature Relations: By engaging the discrete analog of curvature and introducing it in polynomial spaces, this research opens avenues for generalized valuation problems, especially in complex and high-dimensional discretized surfaces.
• Barycentric Refinement: The extension of results to barycentric refinement emphasizes the agility of the approach in handling more intricate structures and predicting emergent properties in deeper subdivisions—as yet another illustration of this theory's extendibility.
• Spectra and Symmetric Functions: The paper positions Gauss-Bonnet and Dehn-Sommerville as gateways to advance spectral graph theory and work with functional symmetries within these domains.

Overall, the paper aligns Dehn-Sommerville with novel insights into manifold symmetries, inferring broader possibilities within computational topology and algebraic combinatorics. Future research may expand into multi-valuate settings like Wu characteristics, leveraging this approach to unpack further connections between topological invariants and symmetric polynomial functions. The research lays the groundwork for potential applications in the optimization of geometric transformations and the analysis of complex network topologies.