The coalgebra of singular chains and the fundamental group

Geometry Topology Seminar
Monday, January 27, 2020 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Manuel Rivera – Purdue University – river227@purdue.edu
Jonathan Beardsley

The goal of this talk is to explain the sense in which the natural algebraic structure of the singular chains on a path-connected space determines its fundamental group functorially. This new basic piece about the algebraic topology of spaces, which tells us that the fundamental group may be determined from homological data, has several interesting and deep implications. An example of a corollary of our statement is the following extension of a classical theorem of Whitehead: a continuous map between path-connected pointed topological spaces is a weak homotopy equivalence if and only if the induced map between the differential graded coalgebras of singular chains is a Koszul weak equivalence (i.e. a quasi-isomorphism after applying the cobar functor). A deeper implication, which is work in progress, is that this allows us to give a complete description of infinity groupoids in terms of homological algebra.

There are three main ingredients that come into play in order to give a precise formulation and proof of our main statement: 1) we extend a classical result of F. Adams from 1956 regarding the “cobar construction” as an algebraic model for the based loop space of a simply connected space, 2) we make use of the homotopical symmetry of the chain approximations to the diagonal map on a space, and 3) we apply a duality theory for algebraic structures known as Koszul duality. This is joint work with Mahmoud Zeinalian and Felix Wierstra.