The fundamental group of a simplicial cocommutative coalgebra

Abstract: In this talk I will describe a functor F from the category of connected simplicial cocommutative coalgebras to differential graded bialgebras satisfying the following properties:

1) if $C$ is the simplicial cocomutative coalgebra of chains on a reduced simplicial set $X$, then the dg bialgebra $F (C )$ is naturally quasi-isomorphic to the chains on the based loop space of $|X|$;

2) $F$ is homotopical in the sense that it sends 'Koszul weak equivalences' (also called '$\Omega$-quasi-isomorphisms') of simplicial cocommutative coalgebras to quasi-isomorphisms of dg bialgebras.

The composition $\Pi_1=G\circ H_0\circ F$, where $H_0$ denotes zero-th homology and $G$ denotes group-like elements, gives rise to a functor from connected simplicial cocomutative coalgebras to the category of groups, which recovers the fundamental group when applied to chains on a simplicial set. We use this construction to extend theorems of Quillen, Sullivan, Mandell, and Goerss to the setting of non-simply connected spaces. The end goal of the program is to provide a complete algebraic (homological) characterization of homotopy types. Some of the results discussed are a joint work with Mahmoud Zeinalian and Felix Wierstra [1].

References:

[1] Manuel Rivera, Felix Wierstra, Mahmoud Zeinalian, The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time, arXiv:2006.05362.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)