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Is there a homology theory that gives a *necessary and sufficient* condition for homotopy eq...

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $ …
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