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Polyhedral products over finite posets

Daisuke Kishimoto, Ran Levi

Abstract

Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $\calp$, that include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $\calp$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that include face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley-Reisner ring of a polyhedral poset and show that, like in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $\calp$ we construct a simplicial poset $s(\calp)$, and show that if $\calp$ is a polyhedral poset then polyhedral products over $\calp$ coincide up to homotopy with the corresponding polyhedral products over $s(\calp)$.

Polyhedral products over finite posets

Abstract

Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset , that include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that include face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley-Reisner ring of a polyhedral poset and show that, like in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset we construct a simplicial poset , and show that if is a polyhedral poset then polyhedral products over coincide up to homotopy with the corresponding polyhedral products over .

Paper Structure

This paper contains 18 sections, 43 theorems, 109 equations.

Table of Contents

  1. Polyhedral products over finite posets
  2. Posets
  3. Polyhedral product
  4. Polyhedral tensor products
  5. Stanley-Reisner ring
  6. Acyclic functors on finite posets
  7. Functor cohomology and the derived functors of the inverse limit
  8. Lower factoring sections
  9. Acyclicity of functors with lower factoring sections
  10. Lower saturated posets
  11. Polyhedral posets
  12. Definition and basic properties
  13. Reduced posets
  14. Stanley-Reisner ring
  15. Colimits and homotopy colimits
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathcal{P}$ be a finite pointed poset, and let $F\colon \mathcal{P}^\textup{op}\to\EuR{Ab}$ be a functor with a lower factoring section $S$ (see Definition lower factoring). Then $F$ is acyclic, i.e. for all $n>0$.

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • ...and 84 more