Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homological algebra and poset versions of the Garland method

Eric Babson, Volkmar Welker

Abstract

Garland introduced a vanishing criterion for a characteristic zero cohomology group of a locally finite and locally connected simplicial complex. The criterion is based on the spectral gaps of the graph Laplacians of the links of faces and has turned out to be effective in a wide range of examples. In this note we extend the approach to include a range of non-simplicial (co)chain complexes associated to combinatorial structures we call Garland posets and elaborate further on the case of cubical complexes.

Homological algebra and poset versions of the Garland method

Abstract

Garland introduced a vanishing criterion for a characteristic zero cohomology group of a locally finite and locally connected simplicial complex. The criterion is based on the spectral gaps of the graph Laplacians of the links of faces and has turned out to be effective in a wide range of examples. In this note we extend the approach to include a range of non-simplicial (co)chain complexes associated to combinatorial structures we call Garland posets and elaborate further on the case of cubical complexes.
Paper Structure (10 sections, 7 theorems, 24 equations, 3 figures)

This paper contains 10 sections, 7 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Garland Structures
  3. Garland Posets
  4. Application to Simplicial and Cubical Complexes
  5. Simplicial
  6. Cubical
  7. Proofs
  8. Examples
  9. Moment angle complexes and branched coverings
  10. Random simplicial and cubical complexes

Key Result

Theorem 2.1

If a Garland structure has $\beta_G < \alpha_G$ then $B$ is exact.

Figures (3)

  • Figure 1: Legend for \ref{['fig:first']} and \ref{['fig:second']}
  • Figure 2: Two pieces of moment-angle complexes $X_K^{\geq 1}$
  • Figure 3: Universal cover of $X_K^{\geq 1}$ for the pieces from \ref{['fig:first']}

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.1
  • proof : Sketch of proof (For a full proof see \ref{['sec:5']}):
  • Corollary 4.1
  • Corollary 4.2
  • ...and 11 more