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

Let R ⊆ Q be a ring with least non-invertible prime p . Let X = X n ∪ α ( ⋃ j ∈ 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 → ⨁ j ∈ J π q ( X n ) R → E ( X R ) → C q → 0 where C q is a subgroup of GL | J | ( R ) × E ( X R n ) . 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.