Adams’ cobar equivalence

Abstract:

Let $F$ be the homotopy fibre of a continuous map $Y\xrightarrow {\omega }X$, with $X$ simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA’s and DGA modules, \[ \Omega {C_{\ast }}(X)\xrightarrow { \simeq }C{U_{\ast }}(\Omega X),\] and \[ \Omega (C_{\ast }^\omega (Y);{C_{\ast }}(X))\xrightarrow { \simeq }C{U_{\ast }}(F),\] where on the left-hand side $\Omega ( - )$ denotes the cobar construction. Our equivalences are natural in $X$ and $\omega$. Using this result we show how to read off the algebra ${H_{\ast }}(\Omega X;R)$ and the ${H_{\ast }}(\Omega X;R)$ module, ${H_{\ast }}(F;R)$, from free models for the singular cochain algebras $C{S^{\ast }}(X)$ and $C{S^{\ast }}(Y)$; here we assume $R$ is a principal ideal domain and $X$ and $Y$ are of finite $R$ type.