Rational homotopy equivalences and singular chains

Abstract

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $\pi_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $\pi_1$-rational homotopy equivalences correspond to maps that induce $\Omega$-quasi-isomorphisms on the rational singular chains, i.e. maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphism and $\Omega$-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative.