On the homotopy classification of spaces by the fixed loop space homology

Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty$) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1,$ a complete homotopy invariant is constructed in terms of the loop space homology $H_*(ΩX).$ This allows us to classify all such $R$-local spaces up to homotopy with a fixed loop space homology.

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