Let $G =(V(G),E(G)))$ be a $n$-regular hypergraph. Let $Z_n$ be a cyclic group with generator $a.$

Definition : A resolution of $G$ is finite partially ordered set $C$ with

  1. $C_0$ is the set of minimal elements coincides with the set$\{(x_1, ..., x_n): \{x_1,...,x_n\} \in E(G)\}$
  2. Action of $Z_n$ is given by $a.(x_1, ..., x_n)= (x_2, ..., x_n , x_1)$ and extend it to a linear action on $C.$

3.Let $x \in C$ and let $x_i = (x_i^1 ,..., x_i^n) \in C_0$ such that $x_i \leq a^i x$ for $1\leq i\leq n$ then $\{x_1^1,x_2^1 ,..., x_n^1\} \in E(G).$

This definition of resolution is totally not clear to me.

Can anyone provide an explicit example to understand this definition?

reference : Igor Kriz , Equivariant cohomology and lower bound to chromatic number, Tran.AMS, 1992.

Another question : I think $C_n \ast C_n \ast \cdots \ast C_n (k \; times \; topological \; join)$ is a $k$-regular hypergraph. Then what is the resolution of it in the above sense? What is the classifying space for this resolution?

Any help will be appreciated .

Thank you.


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