WI-posets, graph complexes and Z_2-equivalences

TL;DR

This paper introduces WI-posets as a new framework to unify the Z_2-homotopy types of various graph complexes, providing new insights and tools for their analysis.

Contribution

It presents WI-posets as an intermediate structure linking graph complexes and Z_2-homotopy types, and applies this to classify complexes and prove new results.

Findings

Finite, free Z_2-complexes are graph complexes.

Determined Z_2-homotopy types of complexes Ind(C_n).

Unified view of graph complexes via WI-posets.

Abstract

We introduce WI-posets as intermediate objects in the study of Z_2-homotopy types of graph complexes. It turns out that (almost) all graph complexes associated to a graph can be viewed as avatars of the same object, as long as their Z_2-homotopy types are concerned. Among the applications are a proof that each finite, free Z_2-complex is a graph complex and an evaluation of Z_2-homotopy types of complexes Ind(C_n) of independence sets in a cycle C_n. The main tools used in the paper are Quillen fiber theorem and Bredon criterion for Z_2-equivalence of Z_2-complexes.