# quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?

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jim -- to rephrase your question: you give the definitions of two things and ask if the difference in ONLY that one definition defines one thing and the other definition defines the other thing. –  algori Jul 4 '10 at 18:04
Andrea- I do have to say that I think it's rather rude to accuse people of being Jim Stasheff impersonators without any evidence. I agree that it would be extraordinarily out of line to post as a well-known mathematician, but since I don't think we've had any serious cases of that, I'd give any poster use what appears to be a IRL name the benefit of the doubt. –  Ben Webster Jul 4 '10 at 19:12
I seem to recall hearing or reading something like: If we restrict to, say, simply connected CW complexes, then X and Y are rationally weak homotopy equivalent (meaning there's a map inducing isomorphisms on homotopy groups tensor Q) if and only if C^*(X;Q) and C^*(Y;Q) are quasi-isomorphic dg algebras. Is this true? Maybe something like this is what Jim is asking about. –  Kevin H. Lin Jul 4 '10 at 20:33
Kevin -- this is certainly true if one considers Sullivan's piecewise polynomial cochains. For singular cochains the "only if" part is obviously true, but the "if" part is subtle (presumably also true, but the details seem to have never appeared in literature). See mathoverflow.net/questions/9146/… –  algori Jul 4 '10 at 20:48
Guys -- let's remove this impersonator nonsense before more people see it. –  algori Jul 6 '10 at 5:08