Auslander-Reiten theory over topological spaces

Auslander-Reiten triangles and quivers are introduced into algebraic topology.It is proved that the existence of Auslander-Reiten triangles characterizes Poincaréduality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant.The theory is applied to spheres whose Auslander-Reiten triangles and quivers arecomputed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out toconsist of $d-1$ copies of ${mathbb Z} A_{infty}$. Hence the quiver is asufficiently sensitive invariant to tell spheres of differentdimension apart.