Self homotopy equivalences and algebraic groups

Speaker: 
Antonio Viruel
Friday, July 26, 2013 - from 16:00 to 16:25
Abstract: 
In a recent work [1], we prove that given any \emph{finite} group $G$ there exist infinitely many rational homotopy types $X_0$ such that the group of self homotopy equivalences of $X_0$ is isomorphic to $G$, that is $G\cong \mathcal E (X_0)$. In this lecture we consider the case of $G$ being infinite. It is well known that given a ``nice'' rational space $X_0$ the group $\mathcal E (X_0)$ is the group of $\mathbb{Q}$-rational points of an algebraic linear group over $\mathbb{Q}$ [2]. Therefore not every infinite group $G$ is isomorphic to the group of self homotopy equivalences of a ``nice'' rational space. We shall find conditions on the minimal Sullivan model of a rational space that ensure the group of self homotopy equivalences being infinite, and will provide non trivial examples of realisation of infinite groups.