New York Journal of Mathematics
Volume 11 (2005) 333-343


Thomas Hüttemann

Total cofibres of diagrams of spectra

Published: July 15, 2005
Keywords: Homotopy limits, homotopy colimits, posets, Bousfield-Kan spectral sequence
Subject: 55P99, 57Q05

If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre Γ (Y) of Y as cofibre(hocolimD (Y) → hocolimC (Y)). We construct a comparison map \hatΓY : holimC Y → Hom (Z, \hatΓ (Y)) to a mapping spectrum of a fibrant replacement of Γ (Y) where Z is a simplicial set obtained from C and D, and characterise those poset pairs D ⊂ C for which \hatΓY is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to Z. For example, if C is a finite polytopal complex with |C| ≅ Bm a ball with boundary sphere |D|, then |Z|≅PL Sm, and \hatΓ(Y) and holimC (Y) agree up to m-fold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for \pi*(Γ(Y)) with E2-term involving higher derived inverse limits of \pi* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.

Author information

Universität Göttingen, Fakultät für Mathematik, Mathematisches Institut, Bunsenstr. 3-5, D-37073 Göttingen, Germany