The Effect of Cell-Attachment on the Group of Self-Equivalences of an Elliptic Space

Abstract

Let X be a simply connected rational elliptic space of formal dimension m, and let $\mathcal{E}(\mathit{X})$ denote the group of homotopy classes of self-equivalences of X. If Y is the space obtained by attaching rational cells of dimension q to X, where $\mathit{q}>2\mathit{m}+1$, then we prove that $\mathcal{E}(\mathit{Y})\cong \mathit{GL}(\mathit{r},\mathbb{Q})\times \mathcal{E}(\mathit{X})$ and ${\mathcal{E}}_{\ast }(\mathit{Y})\cong {\mathcal{E}}_{\ast }(\mathit{X})$, where $\mathit{r}=\dim {\mathit{H}}_{\mathit{q}}(\mathit{Y},\mathbb{Q})$. Here ${\mathcal{E}}_{\ast }(\mathit{X})$ denotes the subgroup of $\mathcal{E}(\mathit{X})$ of the elements inducing the identity on the homology groups. Consequently, we show that, for any finite group G and for any $\mathit{r}\in \mathbb{N}$, there exists a simply connected space X such that $\mathcal{E}(\mathit{X})\cong \mathit{GL}(\mathit{r},\mathbb{Q})\times \mathit{G}$.