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

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 ${\pi }_1(X)={\pi }_1(Y)$.

17 pages. arXiv admin note: text overlap with arXiv:1304.6328

References in corpus (1)

• Cellular coalgebras over the Barratt-Eccles operad I