 

Thomas Hüttemann
Total cofibres of diagrams of spectra


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


Abstract
If Y is a diagram of spectra indexed by an arbitrary poset C together
with a specified subposet D, we define the total cofibre Γ
(Y) of Y as
cofibre(hocolim_{D} (Y) → hocolim_{C} (Y)).
We construct a comparison map \hatΓ_{Y} :
holim_{C} 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 \hatΓ_{Y} 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 ≅ B^{m} a ball with boundary sphere D, then Z≅_{PL}
S^{m}, and \hatΓ(Y) and holim_{C} (Y) agree up to mfold looping and
up to stable equivalence. As an application of the general result we give a
spectral sequence for \pi_{*}(Γ(Y)) with E_{2}term involving higher
derived inverse limits of \pi_{*} (Y), generalising earlier constructions
for spacevalued diagrams indexed by the face lattice of a polytope.


Author information
Universität Göttingen, Fakultät für Mathematik, Mathematisches Institut, Bunsenstr. 35, D37073 Göttingen, Germany
huette@unimath.gwdg.de
http://www.unimath.gwdg.de/huette/

