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Let R be a subring of the rationals with least non-invertible prime p. Let X= X^{n} \cup_{\alpha} (\bigcup_{j \in J} e^{q}) be a cell attachment with Jfinite and q small with respect to p. Let E(X_R) denote the group of homotopyself-equivalences of the R-localization X_R. We use DG Lie models to constructa short exact sequence 0 \to \bigoplus_{j \in J}\pi_q(X^n)_R \to E(X_R) \to C^q\to 0 where C^q is a subgroup of GL_{|J|}(R) \times E(X^n_R). We obtain arelated result for the R-localization of the nilpotent group E_*(X) of classesinducing the identity on homology. We deduce some explicit calculations of bothgroups for spaces with few cells.

The effect of cell-attachment on the group of self-equivalences of an R-localized space