2022-01-17T06:45:16Zhttps://soar-ir.repo.nii.ac.jp/oaioai:soar-ir.repo.nii.ac.jp:000117542021-09-02T05:47:09Z1169:1170On the levels of maps and topological realization of objects in a triangulated categoryKuribayashi, KatsuhikoThe 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 Jorgensen. 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.ArticleJOURNAL OF PURE AND APPLIED ALGEBRA. 216(4):752-765 (2012)journal articleELSEVIER SCIENCE BV2012-04application/pdfJOURNAL OF PURE AND APPLIED ALGEBRA42167527650022-4049AA00705737https://soar-ir.repo.nii.ac.jp/record/11754/files/levels_spaces-neo1.pdfeng10.1016/j.jpaa.2011.08.009https://doi.org/10.1016/j.jpaa.2011.08.009CopyrightÂ© 2011 Elsevier B.V.