JournalscmhVol. 79 , No. 1DOI 10.1007/s00014-001-0795-4

Auslander-Reiten theory over topological spaces

  • Peter Jørgensen

    University of Newcastle, Newcastle upon Tyne, UK
Auslander-Reiten theory over topological spaces cover

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 dd-dimensional sphere turns out to consist of d1d-1 copies of ZA{\mathbb Z} A_{\infty}. Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.