On the classification problem of the quasi-isomorphism classes of free chain algebras

The present paper is devoted to the classification problem of the quasi-isomorphism classes of free differential graded algebras (dgas) over a (P.I.D) R. We introduce the notion of coherent homomorphisms, perfect and quasi-perfect dgas (the Adams–Hilton model of simply connected CW-complex such that H*(X,R) is free is a such a dga) and our first main result asserts that two perfect (quasi-perfect) dgas are quasi-isomorphic if and only if their Whitehead exact sequences are coherently isomorphic. Moreover we define the notion of a strong isomorphism between the Whitehead exact sequences and we show that two free R-dgas, of which their Whitehead exact sequences are strongly isomorphic, are quasi-isomorphic.