Adams' Cobar Equivalence on JSTOR

Transactions of the American Mathematical Society

Description:

This monthly journal, published since 1900, is devoted entirely to research in pure and applied mathematics, and, in general, includes longer papers than those in the Proceedings of the American Mathematical Society.

Coverage: 1900-2013 (Vol. 1, No. 1 - Vol. 365, No. 12)

Moving Wall: 5 years (What is the moving wall?)

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ISSN: 00029947

EISSN: 10886850

Subjects: Mathematics, Science & Mathematics

Collections: Arts & Sciences I Collection, JSTOR Archival Journal & Primary Source Collection, JSTOR Essential Collection, Mathematics & Statistics Collection

Abstract

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

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Adams' Cobar Equivalence

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Abstract