Title: Singular chains and the fundamental group
Speaker: Manuel Rivera (University of Miami)
Abstract: I will explain how the singular chains on a path connected space, considered as a coalgebra with extra algebraic structure, encodes the data of the fundamental group of the space. I will then introduce an algebraic notion of weak equivalence between differential graded coalgebras, which is stronger than quasi-isomorphism, to show the following version of a classical theorem of Whitehead: a map between path connected spaces is a weak homotopy equivalence if and only if the induced map at the level of singular chains is a weak equivalence (in the strong sense) of dg coalgebras. These observations rely on an extension of a classical theorem of Adams relating the cobar construction to the based loop space. Our end goal is to characterize possibly non-simply connected homotopy types through E-infinity coalgebras extending a theorem of Mandell. This is joint work with M. Zeinalian and F. Wierstra.
Seminar URL: https://www.asc.ohio-state.edu/fontes.17/homotopy_seminar/
