On the quasi-isomorphism type of a perfect chain algebra

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 $H_{*}(X,R)$ is free). We show that if $(T(V),\partial)$ and $(T(W),δ)$ are two perfect dgas, then $(T(W),δ)\in \{(T(V),\partial)\}$ 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 $\big((T(V),\widetilde{\partial}),(π_{n})_{n\geq 2}\big)$ consisting with a perfect dga $(T(V),\widetilde{\partial})$ and a family of extensions $(π_{n})_{n\geq 2}$ and we establish that if $(T(W),\widetildeδ)\in \{(T(V),\widetilde{\partial})\}$ and if the extensions $(π_{n})_{n\geq 2}$ and $(π'_{n})_{n\geq 2}$ are isomorphic (in a certain sense), then $(T(W),δ)\in \{(T(V),\partial)\}$.

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