2011/02/16 by Katsuhiko Kuribayashi, Kuribayashi, Katsuhiko
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.AT #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.1102.3271
21 pages. The title is changed
openalex publication_date 2011/02/16 · arxiv created 2011/07/05 · arxiv updated 2011/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover we provide a method to compute the invariant for spaces over a \K-formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander-Reiten theory for spaces due to Jørgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.