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 -dimensional sphere turns out to consist of copies of . Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.
Cite this article
Peter Jørgensen, Auslander-Reiten theory over topological spaces. Comment. Math. Helv. 79 (2004), no. 1, pp. 160–182DOI 10.1007/S00014-001-0795-4