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

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 J finite and q small with respect to p. Let E(X_R) denote the group of homotopy self-equivalences of the R-localization X_R. We use DG Lie models to construct a 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|}® \times E(X^n_R). We obtain a related result for the R-localization of the nilpotent group E_*(X) of classes inducing the identity on homology. We deduce some explicit calculations of both groups for spaces with few cells.