$K(Z,2)$ out of circular permutations

TL;DR

The paper introduces a finite simplicial model for the classifying space of U(1), constructed from circular permutations, which represents minimally triangulated circle bundles and aligns with foundational homotopy theory.

Contribution

It presents a new finite simplicial complex model for K(Z,2) based on circular permutations, connecting combinatorial structures with homotopy-theoretic foundations.

Findings

Finite simplicial model for K(Z,2) from circular permutations

Model represents minimally triangulated circle bundles

Homotopy equivalence with classifying space is canonical

Abstract

We discuss $\pmb{SC}_*$, a simplicial homotopy model of $K(Z,2)$ constructed from circular permutations. In any dimension, the number of simplices in the model is finite. The complex $\pmb{SC}_*$ naturally manifests as a simplicial set representing ``minimally" triangulated circle bundles over simplicial bases. On the other hand, existence of the homotopy equivalence $|\pmb{SC}_*| \approx B(U(1)) \approx K(Z,2)$ appears to be a canonical fact from the foundations of the theory of crossed simplicial groups.