Average Betti numbers of induced subcomplexes in triangulations of manifolds
| Autor: | Giulia Codenotti, Francisco Santos, Jonathan Spreer |
|---|---|
| Přispěvatelé: | Universidad de Cantabria |
| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
Strongly connected component Betti number 500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik Stacked and neighborly spheres 0102 computer and information sciences Type (model theory) 01 natural sciences mu-vector Theoretical Computer Science Combinatorics Mathematics - Geometric Topology Simplicial complex graded Betti numbers FOS: Mathematics Perfect elimination order Mathematics - Combinatorics Discrete Mathematics and Combinatorics σ-vector 0101 mathematics Commutative algebra µ-vector Mathematics stacked and neighborly spheres Ring (mathematics) Mathematics::Commutative Algebra Simplicial complexes Triangulations of manifolds Applied Mathematics 010102 general mathematics triangulations of manifolds Geometric Topology (math.GT) Computational Theory and Mathematics 010201 computation theory & mathematics sigma-vector τ -vector Combinatorics (math.CO) Geometry and Topology implicial complexes Symmetry (geometry) Graded Betti numbers tau-vector 57Q15 05E45 13F55 57M15 Billera-Lee polytopes |
| Zdroj: | The electronic journal of combinatorics 27(3) (2020) UCrea Repositorio Abierto de la Universidad de Cantabria Universidad de Cantabria (UC) |
| DOI: | 10.17169/refubium-28406 |
| Popis: | We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers. Comment: 40 pages, 5 figures. Changes from v1: new title; some proofs shortened, and some omitted. This version has been accepted for publication in The Electronic Journal of Combinatorics |
| Databáze: | OpenAIRE |
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