\(E_{\infty}\)-model for a mapping space
\(E_\infty\)-algebras are good tools for the study of the homotopy theory of topological spaces (see [\textit{M. A. Mandell}, Topology 40, No.1, 43-94 (2001; Zbl 0974.55004)]). Here the authors develop an application of the homotopy theory of \(E_\infty\)-algebras to the study of the mapping space \(X^Y=\text{Top}(Y,X)\). They define a cofibrant model of the commutative operad, denoted \(C_\infty\), in relation with the Barratt-Ecless operad (see [\textit{C. Berger} and \textit{B. Fresse}, Math. Proc. Camb. Philos. Soc. 137, No.1, 135-174 (2004; Zbl 1056.55006)]). Next, the main result is proved: If \(Y_{\bullet}\) is a connected simplicial set of finite type whose geometric realization is \(Y\), to any topological space \(X\) (respectively any \(C_\infty\)-algebra \(A\)) is naturally associated a cosimplicial space, \(X^{Y_{\bullet}}\) (respectively a simplicial \(C_\infty\)-algebra, \(A^{Y_{\bullet}}\)). Then, \textbf{Theorem.} Let \(X\) be a connected space of \(\mathbf{k}\)-finite type. If A is a cofibrant replacement of the \(C_\infty\)-algebra \(N^*X\) then there exists a natural map of \(C_\infty\)-algebras \(NA^{Y_{\bullet}}\to N^*(X^Y)\), which is a quasi-isomorphism when the cosimplicial set \(X^{Y_{\bullet}}\) is convergent. \(N^*\) denotes the normalized singular cochain functor with coefficients in an arbitrary commutative ring \(\mathbf{k}\), and \(N\) is the normalization functor of the total chain complex associated to the simplicial set. When \(Y=S^1\), a natural quasi isomorphism of \(C_\infty\)-algebras from the operadic Hochschild chain complex to the normalized singular cochains on the free loop space is obtained.
scientific article; zbMATH DE number 2136839
scientific article