$\mathfrak{S}$-coalgebras determine fundamental groups

TL;DR

This paper demonstrates that $rak{S}$-coalgebras over the Barratt-Eccles operad uniquely determine the fundamental group of simplicial complexes, linking algebraic structures to topological invariants.

Contribution

It establishes that isomorphisms of $rak{S}$-coalgebras imply equivalences of the 3-skeletons and the fundamental groups of simplicial complexes.

Findings

Isomorphism of $rak{S}$-coalgebras implies 3-skeletons are weakly equivalent.

Fundamental groups of the complexes are equal under coalgebra isomorphism.

The 3-skeleton captures enough information to determine the fundamental group.

Abstract

In this paper, we extend earlier work by showing that if $X$ and $Y$ are simplicial complexes (i.e. simplicial sets whose nondegenerate simplices are determined by their vertices), an isomorphism $\mathcal{C}(X)\cong\mathcal{C}(Y)$ of coalgebras over the Barratt-Eccles operad implies that the 3-skeleton of $X$ is weakly equivalent to the 3-skeleton of $Y$, also implying that $\pi_{1}(X)=\pi_{1}(Y)$.