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Transactions of the American Mathematical Society

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Adams' cobar equivalence

Authors: Yves Félix, Stephen Halperin and Jean-Claude Thomas
Journal: Trans. Amer. Math. Soc. 329 (1992), 531-549
MSC: Primary 55P35; Secondary 55R20, 55T20
MathSciNet review: 1036001
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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,

$\displaystyle \Omega {C_{\ast}}(X)\xrightarrow{ \simeq }C{U_{\ast}}(\Omega X),$


$\displaystyle \Omega (C_{\ast}^\omega (Y);{C_{\ast}}(X))\xrightarrow{ \simeq }C{U_{\ast}}(F),$

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.

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Keywords: Cobar construction, homotopy fiber, free model, Eilenberg-Moore spectral sequence
Article copyright: © Copyright 1992 American Mathematical Society

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