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 $H_{*}(X,R)$ 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 $\big((T(V),\widetilde{\partial}),(\pi_{n})_{n\geq 2}\big)$ consisting with a perfect dga $(T(V),\widetilde{\partial})$ and a family of extensions $(\pi_{n})_{n\geq 2}$ and we establish that if $(T(W),\widetilde{\delta})\in \{(T(V),\widetilde{\partial})\}$ and if the extensions $(\pi_{n})_{n\geq 2}$ and $(\pi'_{n})_{n\geq 2}$ are isomorphic (in a certain sense), then $(T(W),\delta)\in \{(T(V),\partial)\}$.