Total cofibres of diagrams of spectra

Thomas Hüttemann
2005 New York Journal of Mathematics New York J. Math   unpublished
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(hocolim D (Y) E hocolim C (Y)). We construct a comparison map ˆ Γ Y : holim C Y E hom(Z, ˆ Γ(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 whichˆΓwhichˆ whichˆΓ Y is a stable equivalence. The characterisation is given in terms of stable
more » ... erms of stable cohomotopy of spaces related to Z. For example, if C is a finite polytopal complex with |C| ∼ = B m a ball with boundary sphere |D|, then |Z| ∼ =P L S m , andˆΓandˆ andˆΓ(Y) and holim C (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 π * (Γ(Y)) with E 2-term involving higher derived inverse limits of π * (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.