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

Justin R. Smith

Published 2013-07-26, updated 2013-08-12Version 2

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 $\pi_{1}(X)=\pi_{1}(Y)$.