  In this paper, Auslander-Reiten triangles are introduced into algebraic
topology, and it is proved that their existence characterizes Poincare duality
spaces.
  Invariants in the form of quivers are also introduced, and Auslander-Reiten
triangles and quivers over spheres are computed.
  The quiver over the d-dimensional sphere turns out to consist of d-1
components, each isomorphic to ZA_{\infty}. So quivers are sufficiently
sensitive invariants to tell spheres of different dimension apart.
