Cochains and homotopy type (Q2498277)

This paper is an extension the author's earlier paper [Topology 40, 43--94 (2001; Zbl 0974.55004)], where he proved that the functor \(C^*(-;\overline{\mathbb{F}}_p)\) which associates to a nilpotent, \(p\)-complete space its singular cochain complex with coefficients in \(\overline{\mathbb{F}}_p\), regarded as an \(E_\infty\)-algebra, is a full embedding of the homotopy category of such spaces into \(E_\infty\)-cochain complexes. In this paper, the author proves an integral version of this result: two (finite type, nilpotent) spaces \(X\) and \(Y\) are weakly equivalent if and only if \(C^*(X;\mathbb Z)\) and \(C^*(Y;\mathbb Z)\) are weakly equivalent (i.e., quasi-isomorphic) as \(E_\infty\)-algebras. Moreover, \(C^*\) is faithful but not full. However, for every \(E_\infty\) map \(f: C^*(Y) \to C^*(X)\), there is a map \(\varepsilon(f):X \to Y\) inducing the same map on cohomology as \(f\). The general idea of the proof is to use what is already known in the \(p\)-adic case and mix it with rational information to obtain the integral result by an arithmetic square argument. As in the predecessor paper, a crucial step is to study the total derived functor of the right adjoint \(T(-;R)\) of \(C^*(-;R)\). For \(R=\mathbb Q\) or \(\widehat{\mathbb Q}\) (the ring of finite adeles), the unit of the adjunction is rationalization of spaces; for \(R=\mathbb Z\), \(T(C^*(Y))\) is closely related to the free loop space on \(Y\). The paper is very cleanly written and a pleasure to read.