On the quasi-isomorphism type of a perfect chain algebra

Abstract

Let $R$ be a (P.I.D) and let $T(V),\partial )$ be a free $R$-dga. The quasi-isomorphism type of $(T(V),\partial )$ is the set, denoted $\{(T(V),\partial )\}$, of all free dgas which are quasi-isomorphic to $(T(V),\partial )$. In this paper we investigate to characterize and to compute the set $\{(T(V),\partial )\}$ for a new class of free dgas called perfect (a special kind of a perfect dga is the Adams-Hilton model of simply connected CW-complex such that $H_{\ast }(X,R)$ is free). We show that if $(T(V),\partial )$ and $(T(W),\delta )$ are two perfect dgas, then $(T(W),\delta )\in \{(T(V),\partial )\}$ if and only if their Whitehead exact sequences are isomorphic. Moreover we show that every dga $(T(V),\partial )$ can be split to give a pair $((T(V),\tilde{\partial }),({\pi }_n{)}_{n\ge 2})$ consisting with a perfect dga $(T(V),\tilde{\partial })$ and a family of extensions $({\pi }_n{)}_{n\ge 2}$ and we establish that if $(T(W),\tilde{\delta })\in \{(T(V),\tilde{\partial })\}$ and if the extensions $({\pi }_n{)}_{n\ge 2}$ and $({\pi }_n^{\prime }{)}_{n\ge 2}$ are isomorphic (in a certain sense), then $(T(W),\delta )\in \{(T(V),\partial )\}$.

References in corpus (2)

• Realizability of the group of rational self-homotopy equivalences
• Rational self-homotopy equivalences and Whitehead exact sequence