Auslander-Reiten theory over topological spaces

Abstract

Auslander-Reiten triangles and quivers are introduced into algebraic topology. It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré duality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant.

The theory is applied to spheres whose Auslander-Reiten triangles and quivers are computed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out to consist of $d-1$ copies of ${\mathbb Z} A_{\infty}$. Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.